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me mus-914

ESSAYS ON ASYMMETRIES, UNCERTAINTY, AND INVESTMENT
By

Woojin Youn

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Economics

1998

ABSTRACT
ESSAYS 0N ASYMMETRIES, UNCERTAINTY, AND INVESTMENT
By

Woojin Youn

This dissertation is a study of the investment decisions of firms, when
they are constrained by uncertainty and behavioral asymmetries. The
options approach to investment encompasses all the models presented in
each context. Asymmetries are shown to give rise to options that affect the
firm's decision to invest under uncertainty.

Chapter 1 describes a general conceptual framework of option analysis.
After a brief introduction to the financial option theory, applications to
physical investments are presented by working through a simple example.
The basic techniques of valuing options on the investment are discussed.

Chapter 2 is concerned with the effectiveness of tax policy under
uncertainty and irreversible investment expenditures. The focus of analysis
is to investigate how taxes, uncertainty, and sunk costs interact in effecting
the firm’s cost of capital and investment. The cost-of—capital formula is
derived, and is compared with its neoclassical version. We find that sunk
costs and uncertainty may be more important determinants of investment
than taxes. The simulated paths of investment show that the model is
empirically relevant and valid.

Chapter 3 examines how asymmetries in the tax system affect the

firm’s investment decisions under uncertainty and irreversibility. The

asymmetries in the tax system are modeled to include an incomplete loss
offset and a stepwise tax credit. We interpret tax claims as complex
options on the firm's uncertain revenue. We find in this model as well that
uncertainty has a more powerful effect on investment than does taxes. The
model identifies the option effect through which the lack of a loss offset
discourages investment. We find that the conditions for tax neutrality need
some modification under uncertainty and irreversibility.

Chapter 4 deals with investment decisions in the presence of moral-
hazard problems in financial markets. In the model, the financial markets
are characterized by the bankruptcy option of firms, loan guarantees by the
government, and asymmetric information that encourage moral hazard. A
simple option framework shows how moral hazard leads to the
overborrowing problem. The cost-of—capital calculations find that a moral-
hazard-prone economy is likely to be contaminated by risky investments and
the volatility of investment. We also argue that government intervention in

loan markets may sometimes be justified in a specific context.

Copyright by
WOOJIN YOUN
1998

ACKNOWLEDGEMENTS

The contributions of many people around me constitute an integral part
of this dissertation. Without their support and encouragement, I would have
hard time in completing my work.

The greatest debt is, of course, due to Charles L. Ballard, chairman of
my dissertation committee. From the selection of a topic to the final
touches on the product, he has made consistent suggestions and comments,
so that my work was always on the right track. I am grateful to John H.
Goddeeris and Jack Meyer, the other committee members, who advised me
on the various parts of the dissertation. Collaboration with them has been
challenging and rewarding. I would also like to thank the faculty of the
Economics Department at Michigan State University for providing me
excellent education in economics.

l have benefited from discussions and cooperation with my fellow
graduate students. Among them, Soomyung Jang, Sanghyop Lee,
Younghoon Sea, and Min Chang deserve special thanks. They were kind
enough to share with me their insight and knowledge.

I would like to thank my wife, Yuja, and my two daughters, Sooyeon
($00) and Jungyeon (Debbie), for having gone through with me many ups
and downs of my graduate study. Our life in East Lansing would not have
been happier without their support and patience. My deepest gratitude goes
to my father who has made me what I am today by providing encouragement

and financial support. I am grateful to the Korea Institute for Industrial

V

Economics and Trade (KIET) for granting me a leave of absence and financial
support. Finally, I would like to dedicate this dissertation to the memory of

my mother who had always given a blind love to me.

vi

TABLE OF CONTENTS

CHAPTER 1

OPTIONS AND INVESTMENT: A CONCEPTUAL FRAMEWORK

1.1. Analogy Between Financial Options and Real Investment

Decisions

1.2. The Basic Concepts of Option Valuation
1.3. An Application to Real Investment Decisions
1.3.1. Non-Traded Output

1.3.2. Traded Output

CHAPTER 2

UNCERTAINTY, IRREVERSIBILITY, AND TAX POLICY
2.1. Introduction
2.2. Mathematical Preliminaries

2.2.1. Stochastic Processes and Brownian Motion

2.2.2. ltO’s Lemma

2.2.3. Application of ltd's Lemma
2.3. The Basic Model
2.4. Tax-Adjusted Cost of Capital
2.5. Simulation of Investment Behavior

2.6. Summary and Conclusions

vii

IX

12

12

17

23

25

.29

36

36

4O

4O

42

44

.45

55

69

77

CHAPTER 3

TAX ASYMMETRIES, IRREVERSIBILITY, AND THE OPTIMAL
INVESTMENT RULE.

3.1. Introduction
3.2. A Two-Period Decision Model
3.3. Tax Asymmetries and Tax Claims: An Option Interpretation
3.4. The Model of Dynamic Investment Decisions
3.4.1. Reversibility and the Value of the Firm
3.4.2. Irreversibility and the Option to Invest

3.5. Summary and Conclusions

CHAPTER 4
ASYMMETRIC INFORMATION, RISKY INVESTMENTS, AND
FINANCIAL MARKET FRAGILITY..

4.1. Introduction
4.2. Moral Hazard Problems in the Financial Market
4.2.1. Asymmetries, Options, and Risky Investments ..

4.2.2. A Simple Illustration

4.3. Bankruptcy Option, Loan Guarantees, and the Optimal

Investment Rule
4.3.1. The Overborrowing Problem
4.3.2. The Dynamic Model of Investment Decisions

4.4. Financial Market Fragility and Financial Crisis

4.5. Summary and Conclusions

viii

..80

...80

84

95

100

.100

114

120

.. 122

122

125

125

.128

.131

131

.139

153

157

..160

LIST OF TABLES

Table 2.1 - Effect of the Tax and Uncertainty on the Threshold Prices

(Costless Reversibility: E=R=2)

Table 2.2 - Effect of Taxes and Uncertainty on the Threshold Prices

(Costly Reversibility: E=2, R= 1.9)

Table 2.3 - Effect of Taxes and Irreversibility on the Threshold Prices

(Certainty Case: a=0)

Table 2.4 - Effect of Taxes and Irreversibility on the Threshold Prices

(Uncertainty Case: a=0.2)

Table 2.5 - Tax Equivalent of an Increase in Uncertainty

(percent. R=1.9)

Table 3.1 - Uncertainty, Tax Rate and Optimal Price for Investment

(Reversibility Case)

Table 3.2 - Uncertainty, Tax Rate and Optimal Price for Investment

(Irreversibility Case)

Table 4.1 - Borrowing (D) and Threshold Price for Investment (P')

(Risk-Neutral Firms)

Table 4.2 - Borrowing (D) and Threshold Price for Investment (P‘)

(Risk-Averse Firms)

60

.62

.64

65

.67

114

117

.147

147

LIST OF FIGURES

Figure 2.1 - Effect of the Tax and Uncertainty on the Threshold Prices
(Costless Reversibility: E=R=2) 60

Figure 2.2 - Effect of Taxes and Uncertainty on the Threshold Prices
(Costly Reversibility: E=2, R= 1.9) 62

Figure 2.3 - Effect of Taxes and Irreversibility on the Threshold Prices
(Certainty Case: a'=0) 64

Figure 2.4 - Effect of Taxes and Irreversibility on the Threshold Prices
(Uncertainty Case: a=O.2) 65

Figure 2.5 - A Sample Path of Output Price 70

Figure 2.6 - A Simulated Path of Investment
(Costless Reversibility Case) 72

Figure 2.7 - A Simulated Path of Investment
(Costly Reversibility Case) 73

Figure 2.8 - Simulated Paths of Investment
(For Higher Parameter Values) 75

Figure 3.1 - The Tax Rate It). the First-Period Revenue (R1). and

the Value of Waiting (W) 89
Figure 3.2 - Uncertainty (0'), the First-Period Revenue (R1). and

the Value of Waiting (W) 90
Figure 3.3 - The Loss Offset, the First-Period Revenue (3,), and

the Value of Waiting (W) 91
Figure 3.4 - Firm's Revenue and Tax Claim 97

Figure 3.5 - The Tax System, the Output Price, and the Value of
the Firm (a'=0.2) 110

Figure 3.6 - The Tax System, the Output Price, and the Value of
the Firm (a'=O.4) 1 11

Figure 4.1 - Asset Value (X) and the Value of the Firm (V)
(When X is certain) 135

Figure 4. 2- Asset Value (X) and the Value of the Firm (V)

(When X is uncertain).

Figure 4.3 - Asset Value (X) and the Value of the Firm (V)
Figure 4.4 - Output Price (P) and Value of the firm (V)

Figure 4.5 - Uncertainty (0') and Cost-of-Capital Lines (P’)

xi

.136

.138

.143

.151

 

‘1.

Introduction

The study of investment is one of the most controversial areas in
modern economics. No single theory has been accepted as coherent and
satisfactory. The lack of integration in theory, in turn, has led to
contradicting empirical results. Still, the characteristics of investment are
not well understood, making it difficult to predict investment to a reasonably
precise level, and rendering the study of investment tricky and cumbersome.
This offers a reason why investment studies have continued to provide
challenging tasks for both theorists and empirical researchers.

This dissertation is a collection of essays on investment. Each of them
aims at laying a small brick on a huge pile of literature in the field. There
are many perspectives from which investment analysis can be tackled. As
the title of the dissertation implies, it purports to examine a firm's
investment decision problem in a situation, where the firm's capital
decisions are constrained by some types of asymmetries, such as irreversible
investment, and its operating environment is uncertain. Thus, the
asymmetries and uncertainty that firms face in their investment decisions
are essential elements of models to be proposed in each context.

It would be a logical first step to place the relevant literature into a
historical perspeCtive. The overview given below is not intended to be
exhaustive, so the specific literature is relegated to each chapter, where

appropriate. Over the last two decades. economic modeling that builds on

 

‘1

2

microeconomic foundations has pursued actively in almost every field of
economics. We need to pay attention to two important achievements that
have had a tremendous effect on economic analysis: the rational
expectations hypothesis, and the role of rigidities and asymmetries in the
economy. Needless to say, the rational expectations hypothesis has become
an important working assumption for every economist. The role of rigidities
and asymmetries in the economic behavior has also provided a good
economic base on which to explain numerous economic phenomena.
Injecting these two ideas into the standard frictionless and static
expectations-based classical model has generated important dynamic results
in many-economic applications. Its early success can be found in explaining
the effectiveness of monetary policy induced by a nominal wage rigidity
(e.g., Fisher (1977)). or in accounting for exchange rate dynamics
associated with sticky prices in the goods sector (e.g., Dornbusch (1976)).
In investment theory, attempts of this sort actually began in the early
19603. The origin of modern investment theory is the standard neoclassical
theory represented by Jorgenson (1963). which assumes competitive
conditions, and ignores uncertainty and rigidities. Subsequent theories
attempted to extend the standard model by investigating other aspects of
investment such as taxes (e.g., Hall and Jorgenson (1967)). Such models
are considered elegant in their theoretical structure, but miss an important
aspect of reality: they imply instantaneous and costless adjustments of
investment in response to changing economic conditions, whereas the actual

investment behavior is rather gradual. In the standard neoclassical model,

3

the firm’s forward-looking behavior has no role to play, and thus the
expectations are assumed to be static.

Eisner and Stortz (1963) is an initial attempt to introduce adjustment
costs caused by the rigidity in capital expansion in order to develop dynamic
and gradual adjustments of investment. Other important early contributions
include Lucas (1967), Gould (1968), and Treadway (1969). The other
strand of contributions endeavored to integrate the neoclassical theory with
Tobin’s q-theory of investment. Hayashi (1982), for example, constructed a
rigorous rational expectations model of investment to derive the optimal rate
of investment as a function of q. Abel (1982) and Summers (1981)
extended this line of work by highlighting the role of expectations in tax
policy analysis.

Increased uncertainty about economic conditions during the 19703 and
19803 urged capital theorists to turn their attention to modeling uncertainty.
Hartman (1972) and Abel (1983b) extended earlier investment models to a
stochastic setting, augmented by adjustment costs. They argued that high
volatility in the output price tends to raise investment, because the firm's
future profitability increases as uncertainty grows. In their models, a firm
changes its labor-capital ratio in response to price fluctuations, which
causes the marginal revenue product of capital to be convex in the output
price. By Jensen's inequality, the firm's profit is increasing in price
uncertainty. Abel (1983b) also shows that the optimal investment rule,
which equates the marginal value of installed capital, or the marginal q, with

the marginal adjustment cost, can be generalized into a case of uncertainty.

 

L

 

A,

4

Another type of rigidity recognized among economists is that firms are
not able to reverse their investments costlessly. Arrow (1968) first showed
in a deterministic context that, when investment is irreversible, firms invest
under two alternating regimes—positive and zero investment. Research
efforts to combine irreversibility in investment with the firm’s uncertainty
are more recent. Theoretical developments along this line are much
attributed to the advent of the theory of option pricing in financial
economics. The early applications of option theory were limited to the
financing aspects of the firm. Yet its applications have gradually expanded
to capital budgeting and the investment process of the firm. Specifically,
financial economists have increasingly recognized that investment projects
abound with the option components in planning, project timing, expansion,
contraction, etc. (See, for example, Trigeorgis (1996).).

Economic application of the option theory to the investment process
was pioneered by McDonald and Siegel (1986), using the option valuation
technique developed by Black and Scholes (1973). They show that, if an
investment is irreversible, there is an opportunity cost to investing today
under uncertainty. This finding implies that uncertainty in the output price

lowers investment, contradicting the pro-investment result mentioned

before. There have since been burgeoning studies on investment that extend

the option-based approach by allowing for more complexities. These
developments are well documented in two separate surveys of Pindyck
(1991) and Dixit (1992). Their co-authored book, Dixit and Pindyck (1994),
also provides a systematic treatment of the new paradigm in investment

theory. The research programs to explore new theoretical possibilities are

 

 

"s

5

still underway ( See, for example, Abel and Eberly (1994, 1996), and Abel,
et al. (1996).).

The new view of investment has seriously altered the conventional
wisdom on investment. The conventional approach to investment employs a
basic principle that, if the net present value (NPV) of the project is greater
than zero, firms should undertake it. This simple NPV rule, however, is
based on an unrealistic premise that investment decisions are costlessly
reversed or, if not, they are once-and-for-all opportunities. In reality, we can
seldom observe an investment that meets this strict requirement.
Investments usually incur expenditures that are at least partially
irrecoverable, and thus are sunk. Firms can delay their investment decision,
in an attempt to capture a better timing to maximize the expected profit of
the project, as uncertainty resolves over time. When the investment is not
completely irreversible, firms can also abandon the project. Thus, they have
the flexibility of delaying and abandoning the investment, or, equivalently,
have the option to invest and to abandon. These options have value, when
the profitability of the project is uncertain.

Once the firm invests, it loses the option to invest, while gaining the
option to abandon. The value of options, therefore, must be added to, and
subtracted from, the standard net present value. The option values are
. adjustment costs of a different sort, that should be distinguished from those
used in the traditional sense. As we will see, the implication for the optimal
rule of investment is that firms do not invest until prices rise substantially,
and do not disinvest until prices fall considerably. Dixit (1997) succinctly

expressed these ideas as follows:

6

“Adjustment costs bring dynamics to a firm's factor demands. When
these costs are linear—proportional to the amount of adjustment—or
lump sum, the dynamics takes the form of intermittent action. No
adjustment is made unless its marginal value is sufficiently high;
therefore there is a range of inaction in the firm's decision rule.
Uncertainty enlarges this range, because of the possibility that future
developments may render the adjustment unnecessary; this can be
expressed as the option value of the status quo." (p.1)

The basic message of the option view of investment is clear:
investment decision that is made 3 on the basis of the naive NPV rule is not
necessarily an optimal, because it ignores options embedded in almost every
investment project. Economists have increasingly taken the view seriously
that committing resources to an investment involves options that have
substantial values. The new approach to investment proves useful in
resolving some empirical puzzles. In actual business decisions, it is not
uncommon to require that a project should yield the expect rate of return
much higher than the usual cost of capital. Summers (1987), for example,
reports that firms are using the threshold rates of return with an average of
17 percent, although he calculated a nominal cost of capital taken from a
survey at a time when expectations of inflation may have been fairly high.
McCauley and Zimmer (1989) estimate the cost of capital for equipment and
machinery in the U.S. at 9.1 to 13.5 percent for the period 1977-1988. The
literature on irreversibility investment demonstrates that the value of the
option to invest is substantial, so that the required rate of return could far
exceed the level that the standard theory suggests. For instance.
simulations performed in McDonald and Siegel (1986) show that projects

should be implemented, only if their present value is twice as much as the

investment costs. The neoclassical investment theory has also failed to

‘u

 

7

provide good empirical relationships to explain observed behaviors of
aggregate investment. Aggregate econometric models do not fit the actual
investment data well. Also, Tobin's q, or the cost of capital measures
usually do not have strong explanatory power in explaining investment
behavior. The recent alternative models move in the direction of offering a
convincing interpretation of the empirical evidence by employing an option
framework of analysis (e.g., Bertola and Caballero (1994)).

Theories of irreversible investment emphasize the importance of sunk
costs in the acquisition and disposition of capital. It is argued that even
small sunk costs significantly affect the firm’s decision to invest, even when
uncertainty is moderate. In the traditional investment theory, uncertainty
can be accommodated by adding a risk premium to the discount rates, if it is
associated with systematic (non-diversifiable) risk. Risk-averse firms then
discount the project’s cash flows heavily, and behave in a cautious manner,
when they make investment decisions. Craine (1989), and Bernheim and
Shoven (1989) are attempts to employ a risk-premium approach. In the
option-based theory, the role of uncertainty is far more fundamental,
regardless of whether it is systematic. Risk-neutral as well as risk-averse
firms can be severely influenced by the options associated with any type of
uncertainty. This is why uncertainty, combined with irreversibility, takes a
central position in the new theory, apart from other key determinants of
investment.

The preceding discussion is confined to the firm's option to invest
resulting from irreversible investment. Yet firms abound with options, so

long as they are able to get better information and/or more resilience. When

 

8

they are constrained by imperfect information, and systemic rigidities and
asymmetries, firms react by gathering more information and enhancing
flexibility. Information and resilience are given exogenously, or can be
created by firms themselves. Many types of R&D investments are well
suited for the latter case. They reveal useful information for firms to decide
whether to undertake the follow-on investments that commercialize
technologies therein. This implies that immediate investing in R&D, rather
than waiting, may bring more valuable options to firms. We can think of
many other contexts for firms to come up with growth options that increase
firm value; see Kester (1984) for more discussion. Again, the option theory
provides a good analytical tool to assess the correct value of the options
involved.

This dissertation studies how the firms’ investment decisions are made
under uncertainty, when the firms are constrained by certain types of
asymmetries. We assume that uncertainty is characterized by a specific
stochastic process in the output price. However, asymmetries are different
across models. In the model of Chapter 2, we assume a typical asymmetry
that the investment is partially irreversible. In the model of Chapter 3, we
employ an additional asymmetry in the tax function. More specifically, the
tax system is characterized by the lack of loss offset, and a ceiling on the
investment tax credit. In the model of Chapter 4, asymmetries are present
in the firm's operating decisions: the firm can temporarily go bankrupt, and
receive a'loan guarantee from the government, if the profits turn out to be

unfavorable. The consistent theme throughout the models is the option

9

approach to investment. Against this general background, we present a
brief description of each chapter contained.

Chapter 1, ”Options and Investment: A Conceptual Framework," sets
the scene for the rest of chapters by providing basic concepts and ideas
about the options approach to investment. After a brief introduction to the
financial option theory, we proceed with an application of the theory to
physical investment, by working through a simple example. We present a
method of valuing the options embedded in the investment, and show the
equivalence of two solution techniques, namely, dynamic programming and
option valuation methods. I

Chapter 2, “Uncertainty, Irreversibility, and Tax Policy," is intended to
answer the following fundamental question: does tax policy matter in
stimulating investment? The effect of taxes on investment is one of the
most controversial area in economics. To address this problem under
uncertainty and irreversibility, we build a small model of a discrete
investment with the parameterization of taxes, uncertainty and
irreversibility. The model allows for flexibility in the firm's sunk costs of
investment in order to show how investment is affected by the degree of
irreversibility. The cost-of-capital formula derived from the model contains
the three main parameters above, as well as others. The option
interpretation of this formula is given in the context of the neoclassical
investment theory. We also perform a sensitivity test to compare the
effects of each parameter on the cost of capital to answer the question

asked above. Finally, we simulate the behavior of investment, and assess

10

the response of investment to each parameter. Simulations will show that
the model is empirically plausible.

Chapter 3, ”Tax Asymmetries, Irreversibility, and the Optimal
Investment Rule," examines how asymmetries in the tax system affect the
firm's investment decisions under uncertainty and irreversibility. The model
incorporates two representative tax asymmetries: an incomplete loss offset,
and a stepwise tax credit. We interpret tax claims as options the tax
authorities have on the firm’s uncertain revenue. Options are shown to vary,
depending on which range of the revenue the firm is placed in. The option
interpretation of the model proves intuitively very useful, and has a direct
bearing on the neoclassical theory. From the solution and simulations of the
model, we contend that uncertainty has a more powerful effect on
investment than taxes, which is consistent with Chapter 2. We point out
that the lack of a loss offset discourages investment, through the option
effect stemming from uncertainty, irreversibility, and tax asymmetries
combined. This conclusion contrasts with that of other models that
emphasize the market risk-sharing role of taxes (e.g., Gordon (1985)). We
reexamine the condition for neutrality of the tax system under uncertainty
and irreversibility. and suggest a qualification for its application.

Chapter 4, ”Asymmetric Information, Risky Investments, and Financial
Market Fragility," deals with the effects of a moral-hazard problem on
investment in financial markets. We employ the same analytical framework
as in the previous chapters, since the model is characterized by asymmetries
and uncertainty. The chapter starts with a discussion of asymmetric

information and moral hazard in financial markets. We show, in a simple

11

option framework, how moral hazard leads to the overborrowing problem.
Using again our dynamic model, we examine the investment decisions of
firms, when they have the bankruptcy option and bank loans with repayment
guarantees by the government. We show that a moral-hazard-prone
economy is likely to be contaminated by risky investments and investment
volatility. The role of asymmetric information and the risk attitudes of firms
are scrutinized. We also argue that government intervention in loan markets

may sometimes be justified in a specific context.

Chapter 1

Options and Investment: A Conceptual Framework

1.1. Analogy Between Financial Options and Real Investment
Decisions

In the financial market, an option gives its owner the right—not the
obligation—to buy or sell assets at a predetermined (exercise) price. A call
option gives the right to buy, while a put option gives the right to sell.

When the owner of an option buys or sells stock, it is called exercising the
option. We can also distinguish options by the timing of exercising. An
American option can be exercised at any time prior to expiry, whereas a
European option can only be exercised at expiry.

When an investment is characterized by irreversibility, the investment
opportunity is equivalent to a financial call option.1 If the firm’s planning
horizon is infinite, a commitment to an investment is like exercising an
American call option with no expiry. A firm must decide whether to invest
now or later in exchange for an irreversible expenditure. To make a term-by-
term analogy, the investment opportunity available in the future is a financial
option, the project to be undertaken is an associated financial asset, and the

investment expenditure is the exercise price of the option. Once the call

 

' By an irreversible investment, we mean that the investment expenditure is sunk once
spent. Irreversibility stems from various reasons, such as the firm- or industry-specific
nature of capital, and the lemons problem in the sales market of capital. For simplicity,
we assume complete irreversibility here. We will see below how incomplete
irreversibility alters the options of the firm.

12

13

option is bought, the buyer cannot resell it to the seller. Likewise, once a
firm proceeds with the investment, it cannot reverse its decision.

An investment opportunity—or the option to invest—has value for the
same reason as a financial option does. The option gives the holder a
flexibility to optimize the timing of exercise. Let us take a stock option as
an example. The price of the stock fluctuates, and is uncertain. When the
holders of an option buy stock today by exercising the option, they will lose
the opportunity to wait until tomorrow or a later date. This opportunity has
value; if the stock price turns out to be higher than the exercise price, they
can make a profit by buying stock. If the stock price turns out to be lower
than the exercise price, they can avoid a loss by holding the option. By the
same reasoning, the option to invest has value. The profitability of a project
is uncertain, so firms can benefit from choosing to wait, and holding the
option to invest. The option to invest is valuable since firms can capture

upside gains, while eliminating downside losses; if the project turns out to
be profitable, they will make a profit by investing. Otherwise, they will
avoid a loss by withholding the investment. When waiting is costless, the
investment will never be made because it always pays to wait. But waiting
is not a free lunch; firms should give away cash flows from the investment
that might be available over the period of waiting. If the cost of waiting is
large enough to offset its benefit, the firm will initiate the investment. This
is the basic thrust of the options approach to irreversible investment.

A firm with a flexible system has a lot of options on various aspects of
investment. As discussed above, the option to invest, which is a call option,

is valuable, when the investment is irreversible. This option disappears,

14

when the investment is made. When the investment is reversible, the firm
gains another option that has value, namely, the option to abandon the
project. The option to abandon is a put option available after the investment
is undertaken, because it is exercised by selling the capital. When the
irreversibility is less than perfect, the value of the firm alters by investing
immediately; it will decrease by losing the option to invest, and will increase
by gaining the option to abandon. The higher degree of irreversibility
increases the value of the lost option to invest, while decreasing the value
of the option to abandon that is gained. If the irreversibility is more severe,
therefore, the firm is less willing to invest immediately, because the value to
be lost would be greater than the value to be gained by investing.

The firm is also constrained by costly expandability as well as costly
reversibility. Costly expandability arises because the firm may in the future
buy capital at a price higher than the current level. In this case, waiting has
an additional cost, since the investment expenditure is expected to increase.
Waiting, or holding the option to invest, will reduce the firm value,
increasing the incentive to invest today. Many options on investment are
also associated with operating flexibility. A good example is the option to
shut down the production operation temporarily. The firm will continue the
operation, if the production line makes a positive profit; otherwise, it will
cease production temporarily, and wait until the operation turns profitable.
This operating option is another source of firm value. In effect, modern
firms have made great strides in increasingly employing the flexible

manufacturing system (FMS) of various sorts, substantially enhancing their

 

15

capital values. Usually, many types of options discussed so far intermingle
in any investment, and affect the value of the firm?

In the neoclassical investment theory, firms have few options
associated with investment. This is because underlying the theory is the
perfectly competitive and frictionless Arrow-Debreu world. The neoclassical
investment theorists took this approach, because it provides a rationale for
abstracting from complications caused by financial considerations of the
firm.3 However, this simplified assumption results in two flaws that seem to
be critical in real-world analysis: static expectations and the lack of options
to the firm. In the neoclassical world, where investment is always reversible
without any cost, the firm's expectations about the prospect for its profits
are not important; the firm can adjust instantaneously and costlessly to its
optimal path of capital stock, whenever expectations turn out to be
incorrect, so that its capital stock deviates from the optimal level. Also
costless reversibility and expandability in investment provide no role for
options to play in determining the level of investment. The new view of
investment, on the other hand, will attach a great importance both to

expectations and options, since it rests totally on different assumptions.

 

2 The fact that the value of the options stems from flexibility implies that the option
model has a wide range of economic and non-economic applications. The applications
in economics range from wage bargaining to currency fluctuations. In finance,
managerial operating flexibilities in capital budgeting are dubbed real options, as
distinguished from financial options. See Dixit and Pindyck (1994), Chap. 1, for more
discussion.

3 The neoclassical theory builds on the Modigliani-Miller (MM) theorem proposed in
Modigliani and Miller (1958). The MM theorem argues that real economic decisions are
independent of a firm's financial structure. In Chapter 4, we will see that the theorem
fails to hold in the presence of asymmetries and options in financial markets.

16
Here is a simple illustration of how options work, even under a world of
certainty. Consider a firm which plans an investment with the following net

present value (NPV) of entire project evaluated at each future point in time:

 

Year 0 1 2 3 4 5

 

Net Present Value -50 -20 30 37 21 18

(in thousands of dollars)

 

Assume now that the firm has perfect information about the NPV to be
earned. Let us first discuss the reversibility case. The firm need not delay
investment so long as the NPV is positive, which implies that it may invest
in Year 2. In Year 3, the firm can retrieve costlessly its investment in Year
2, and reinvest. Hence investing in Year 2 and Year 3 does not make any

difference.

This is not the case, if reversibility is costly. It is not optimal for the
firm to invest in Year 2, since reversing the investment in Year 3 is costly.
So the firm will wait and invest in Year 3. Irreversibility makes a difference
even if there is no uncertainty. It is, therefore, optimal for the firm to pin
down the best timing of investment along the time horizon. This implies that
a positive net present value (NPV) does not mean that the project should be
undertaken, nor does a negative NPV mean that it should be abandoned
immediately. The optimal point to invest is Year 3. Waiting, therefore, has

value equal to the difference between the two NPVs at Year 3 and Year 0:

17

$87,000= $37,000-(-$50,000). This value comes from the firm’s flexibility

of delaying investment in response to irreversibility.

1.2. The Basic Concepts of Option Valuation

The valuation of an option is based on the arbitrage principle in financial
economics.4 Its basic principle—widely known as contingent claims
analysis (CCAl—is associated with constructing a risk-hedging portfolio. To
show how to do this in the European option case. let S be the current price
of the underlying asset (a stock or a project under consideration). T be the
time to expiry, and X be the given exercise price, respectively. The call and
the put options have the same exercise price and expiration date.

Intuitively, the option value is affected by the expected value. of S, which, in
turn, depends on the current price of S, and the time T. Thus, we can
denote the value of a call option by a functional relationship, 6/8, T; X), and
the value of a put option by PIS, T; X). Consider first the value of the call at
expiry, T=0. If S>X, the option will be exercised with a payoff equal to S-

X; otherwise the option will expire worthless. This can be written as
0/8, 0; X)=Max[S-X, 0]. (1)
By the same reasoning, the value of the put at expiry can be expressed as

PIS, 0; X)=Max[X-S, 0]. (2)

 

‘ The valuation of options had been an intriguing but elusive question to financial
economists for many years, before Fisher Black and Myron Scholes made a major
breakthrough in their 1973 Journal of Political Economy paper. The option pricing
model has since become an integral part of every finance textbook. See Brealey and
Myers (1991), Chaps. 20 and 21, and Copeland, at al. (1991), Chap. 15, for a good
introductory discussion. Wilmott, et al. (1995) offers a more rigorous treatment.

18

Consider now a portfolio consisting of holding long one stock and one
put option, and holding short one call option. Then the value of the portfolio

is given by
F=S+P-C (3)

The payoff for this portfolio at expiry is

S +Max[X—S, 0]-Max[S-X, 0], (4)
which can be written as

S + lX-Sl-O =X if 896 (5’)

S+0-(S-Xl=X if S>X (5")

The payoff is always X, regardless of the value of the stock at expiry.
Notice that the portfolio's risk has been eliminated by hedging. Since this
payoff is given at expiry lT=0/. the next step is to value the portfolio prior
to expiry (T: t, t>0). Assuming that the time is continuous, the portfolio
should be worth Xe'", where r is the riskless interest rate, because the
return from the riskless portfolio must equal the return from borrowing
money that gives a sure amount of $X at expiry. If not, an arbitrage
opportunity exists by a proper combination of buying and selling options and
stocks, and borrowing or lending. The no-arbitrage condition stipulates a

fundamental relationship between the asset values at time T=t,

S+P-C=Xe'". (6)

 

19

This relationship is called the put-call parity, and serves as the basis of
option valuation.5 As we have discussed, undertaking a (completely)
irreversible investment is analogous to exercising an American call option.
Assuming that waiting is costless. we can value the call option at time T=t

as
C = S-Xe’", (7)

since complete irreversibility rules out the put option.“ The value of the call
option equals that of a portfolio consisting of buying one unit of the stock,
and at the same time borrowing money equal to the present value of the
exercise price.

We have so far discussed the option valuation in a world of perfect
certainty, where investors have perfect information about the future prices
of assets, and the return to all assets must be equal in equilibrium. In a
world of uncertainty, the extended version of expression (7), widely known
as the Black-Scholes formula, is used for option valuation. The formula
contains the probability terms associated with the value of the underlying
asset at expiry. We do not attempt to get into the Black-Scholes model

because it is beyond our purpose: see Wilmott (1995) for a treatment.’

 

5 This relationship is valid only for European options, since the values of American
options may be different from those of European options due to early exercise.

6 When an American call option is for a dividend-paying stock, waiting incurs the cost
of giving away the receipt of dividend payments. Accordingly, we are discussing how
to value an American call option without dividends. American and European call
options have the same value, if they are associated with non-dividend-paying stocks.
Valuing an American option with dividends is a little tricky, because the cost of waiting
should be considered.

7 The derivation of the Black-Scholes formula is cumbersome, involving the solution of
a partial differential equation. It is used to value European call (put) options, and
American call options with no dividends.

20

Instead. we will work through a two-period example to get an intuitive
understanding of the idea.

As a first step, let us denote the current stock price by S, and the value
of a call option by C. S is assumed to go either up to S, with probability 0,
or down to S, with probability (I-q) over the next period. The movement of

S and C can be depicted as

q/ S”; C” = M8XISU-X, 0]

S

1'q 84; Cd=MaX[$d-X, 0].

The Black-Scholes formula is based on the principle that we can hedge the
risk in a portfolio by taking opposite positions in the underlying stock and
the call option since they are perfectly correlated. This is equivalent to
saying that the call value is replicated by a hypothetical portfolio consisting
(of buying N shares of the stock, and borrowing $8 worth of money at the
risk-free interest rate. This is called a replicating strategy. Since the
portfolio would exactly replicate the payoffs of the call option in any state of

nature, their current values are also equal, such that
C=NS-B. (8)
In the following period, this relationship is expressed as

C,,=NS,,-{1+rIB, (9)

and

21

Cd=NSd-(7+r)8. (10)

Solving these two equations for N and 8 gives

N=£¢LL§1, (11)
Su -8d
and
B=____Ns,—c,. (12)
1+r

N is the amount of the stock required to replicate one option, and is called
the option delta or hedge ratio. Substituting equations (11) and (12) back
into equation (8) gives a useful formula to value the call option in terms of

S, 8,, 8,,r, and X,

C: pc, +rr-p1c,’ where p=11+rIS-Sd.

(13)
1+r Su—Sd

 

 

This formula has a remarkable feature. The probability q and the investors'
attitudes toward risk are irrelevant to the formula. In equilibrium in the
asset market, these factor alters the call value indirectly through their effect
on S, S,,, 8,), and r. The attitudes toward risk do not matter, because every
investor will take advantage of riskless arbitrage opportunities.‘3 To see this

intuitively, we can rearrange equation (8) as

NS-C=B. (14)

 

a In some circumstances, where the underlying asset is not financial assets traded in
the market, for example, risk preference affects the value of an option. We can
accommodate this situation using an appropriate framework (e.g., capital asset pricing
model (CAPM)). See Trigeorgis (1994), Chap. 3, for a more discussion.

22

which implies that a portfolio consisting of buying N shares of stock and
selling short one call option replicates in the next period a sure amount of
money equal to $(1 +r)B. If the equality does not hold, arbitrageurs
intervene, and profit opportunities will be arbitraged away. This suggests an
alternative way of valuing an option. We can pretend that investors are risk-
neutral, so that all the assets would earn the same risk-free rate of return in
equilibrium. We then discount the expected value of the option at the risk-
free rate to get the current value.

To do this, let R, and R, be the rate of return on the stock in each state,
p be the probability that corresponds to q in the risk-neutral .world, and r be
the risk-free rate. In equilibrium, the stock’s expected rate of return equals

the riskless rate of return,
pR,,+(1-led=r, (15)

where Ru=(S,/S)-1 and R,=IS,/Sl-1.

Solving for the risk-neutral probability p, gives

p=_lid_=/1+rIS-Sd. (15)
RU-Rd Sir—8d

 

The current value of the option is equal to the expected payoff of the option.

discounted at the rate r,

(.‘= p0,, +(1-pICd.
1+r

(17)

 

The solutions for p and C are exactly the same as we derived in (13).

23

1.3. An Application to Real Investment Decisions

Before we proceed with examples of how to apply the option valuation
technique (OVT) to a real investment, it would be helpful to discuss its
advantage over two other conventional techniques: the net-present-value
method (NPV) and the decision-tree analysis (DTA). The options approach to
investment is very flexible, because it encompasses the other two methods
of project evaluation: The OVT draws from the NPV a technique of finding a
risk-comparable portfolio, while borrowing from the DTA an idea of valuing
flexibility.

The NPV method first determines the expected net cash flows that a
project will generate. and then discounts them with a proper rate that allows
for risk inherent in the project. A typical method of determining the discount
rate is to search for a hypothetical traded asset or a portfolio with the
equivalent risk. The NPV method, however, does not allow for any
contingencies. The initial decision to accept or reject the project is not
subject to change over its entire life. The DTA, on the other hand, takes
explicitly into account the interdependencies between a series of decisions.
The firm's prior decisions are reevaluated at a time when new information is
available. Clearly, the DTA is better than the NPV in order to reach an
optimal decision over time. The problem with the DTA is that finding the
appropriate risk-adjusted discount rate is very tricky, because the risk of the
project alters relentlessly as contingencies affecting the project value evolve
over time. As we shall see later, the OVT proves to be well suited to

address both the firm’s operating flexibilities and risk adjustments.

24

Finally, the fact that a typical firm has a large pool of assets brings up
an issue that deserves some attention. In valuing the firm as a whole, it is
important to ask the following question beforehand: Is it justified to use the
arithmetic sum of individual asset values to value the entire firm? Since
diversification raises the firm value, a portfolio of assets may have a greater
value than does the sum of the values of its components. If so, we may be
faced by a daunting task in order to value the firm; if a diversified firm has
more value, each asset has to be valued for its contribution to the firm’s
portfolio of assets, not as an independent component. Fortunately, a very
general concept in corporate finance comes to the rescue, namely, the
principle of value additivity. The principle argues that, if the capital market
is perfect, and investors can diversify their assets on their own, a firm’s
diversification is redundant. and does not affect the value of the firm. That
is the basis on which we can value the firm without worrying about the
validity of an adding-up of individual asset values. See Brealey and Myers
(1991) for a discussion of the practical importance of value additivity in
various contexts.

Let us now go on to the illustrative examples. The discussion will
follow for two polar situations, depending on whether the firm’s output is
traded. As we shall see. different replication strategies are used between
these two cases. The non-traded case will discuss three sub-cases,
depending on whether delaying an investment is available and whether
delaying has a cost. This treatment is helpful in comparing the three

different approaches of project evaluation discussed above.

25

1.3.1 . Non-Traded Output
1:hn in'n vill r nin

Consider a firm’s problem of investing in a single new project over the
next two periods. Suppose that that: A) the firm cannot delay the
investment, so the investment decision should made in the first period; 8)
the project yields in the second period a revenue equal to either V,,= $150 or
V,,= $100, with an equal probability q=0.5; and C) the project requires an
initial expenditure equal to K= $110.

Since this project is completely new to the firm, we may not use the
firm's (weighted) average cost of capital (WACC). The standard net-present-
value (NPV) method tells us that, in order to determine the correct discount
rate, we must find an asset or a portfolio that has the same risk
characteristics with the project under consideration, and is currently traded.
Suppose that this replicating asset has the expected payoffs equal to Su=30
and Sd=20 in each state, and currently priced at 8: $21. Note that the
asset's payoff is proportional to that of the project: V,/S,,= V/S,,=5. Hence
we can use the discount rate of the asset, because they have the same risk.

Denoting the discount rate by p. the standard NPV formula is written as

S= qSu +(1-qISd .

 

(18)
1+ p
Solving for p and plugging in the associated numbers yields
p=qs"+"-qlsd-1=O.19. (19)

 

S

26
This is the discount rate we are seeking. The NPV of the project is then

computed according to

= 0V, +II—qIV,
1+p

 

NPVS ~K=$105-$110=$-5. (20)

The simple NPV method recommends that the project should be rejected.
Let us now value the project using the option valuation technique
(OVT). It is intuitively obvious that the OVT will produce the same result,
since there is no option embedded in the project. As we have seen in the
last section, the OVT uses the hypothetical risk-neutral probability p, in
place of the actual probability q, in valuing an asset. Assuming the risk-free

interest rate r=0.1, p can be computed from equation (16).

=(7+f}S-Sd =o.31. (21)
Su —Sd
Using p as the weight probability, the net present value of the project is

pV,,+(1—pr,
7+r

 

NPV,= —K=$105-$110=$-5. (22)

This is exactly the same value as we have computed in equation (20).

I i i v il w' h
When a delay of the investment is possible. the firm must extend its
planning horizon. and decide the optimal timing of investment over the two
periods. In the diagram below, E, and E, denote the net payoffs from the
optimal investment decision in the second period; it earns a net payoff of

$29 by investing in the good state, while it earns nothing by giving up

 

27

investing in bad state. V,= $105 is the expected gross payoff from the
investment undertaken in first period as in equation (20). when the
investment is made in the first period. The firm first calculates the expected
net payoff from the optimal decision in the second period, and then
compares it with the expected net payoff from the optimal decision in the

first period to decide whether and when to invest.

WK; $150;E,,=Max[V,,-l1 +r)K.0]= $29
V,=$105, E,=?
(K=$110) \
1-q=0.5 V,=$100;E,=Max[V,-(1 +r)K,0]=$0.

In effect, this is exactly what the decision-tree analysis (DTA) asks us to do.
In a more complicated case, the DTA enables us to work through all the
decision nodes to reach a best solution. To compute the expected payoff
from the optimal decision in the second period, we determine the appropriate
discount rate. Note that the discount rate derived in the Jorgensonian case
above (p=0.19) is not valid, since the newly characterized investment
opportunity is no longer correlated with the replicating portfolio used there.
Unfortunately, there is no way the DTA can solve this problem. Hence we
ought to find a replicating portfolio using the NPV, and get a risk-neutral

probability p. In our case, p is already computed in equation (21) as

___ (1+r/S-S,
su—Sd

=O.31. (23)

 

28

It is now obvious why the OVT combines the principles of the DTA and
the NPV. The present value of the investment opportunity is calculated as

the weighted value of E, and E,, discounted by the riskless interest rate r,

pEu + (1 —p)E,

NPV": 1+r

=$8.2. (24)

 

The OVT suggests that the project should not be rejected. The example
shows that the value of a project can be enhanced by waiting and making an
investment decision in the second period. The difference between NPV, in
equation (24) and NPV, in equation (20) is a value addition achieved by

waiting,
V,=NPV,-NPV,=$8.2-($-5)=$13.2. (25)

The basic principle of the option valuation enables us to value the flexibility

in the firm's investment decisions.

:Wn Iini v'llwih
We now introduce a cost of waiting in our example. Suppose that. if
the firm were to wait, it would lose in the current period a value equal to a

constant proportion 6 of the value of the project.9 Assuming that 6:0.1,

the expected payoffs of the second period would be

E,=Max[(1-8)V,-(1+rIK, 0]=$18.7. (26)
and

E,=Max[(1-6) V,-(7 +rlK, 0]: $0. (27)

 

9 The lost valued is assumed, for simplicity, to be exogenous. Needless to say, we can
endogenlze it by rendering it dependent on other variables.

 

29

Working through the same steps from (23) through (25). we obtain

 

 

Su -Sd
NPv,=pE"+"’p’Ed =$5.3, (29)
1+r
and
V,=NPV,-NPV,=$5.3-($-5)=$10.3. (30)

In this case, waiting is still optimal, although the value of waiting has
decreased. In other cases where the cost of waiting is sufficiently large, it

would be desirable to undertake an investment in the first period.10

1.3.2. Traded Output‘1

When a firm’s project produces an output that is traded in the market,
an investment opportunity—that is, an option to invest in the future— is
perfectly correlated with the output price. Hence the firm's output is the
underlying asset of the option to invest in the future. so a risk-hedging
portfolio can be constructed using the option and the output without a need
to find another twin asset. Consider a firm that is planning a single project
that is irreversible. Assume that the project produces one unit of output
each period indefinitely, and the initial investment expenditure, K= $500, is
the only cost. Suppose that the current price of output is Po: $80, but it

will either rise to P,= $120 or fall to P,= $40 with equal probability q=0.5

 

1° For example, we can calculate the critical value of 8=0.31 at which V,=0. Doing an
investment immediately is optimal, when 6 exceeds that level.
'1 The example in this section is based on Dixit and Pindyck (1994), Chap. 2.

30

over the next period, and will remain the same forever. We assume that the
firm behaves as if it were risk-neutral by eliminating the market risk of the
project through diversification.12

As in the last example. let us solve the firm’s optimal investment
decision problem under different assumptions. The first solution does not
allow for waiting. Letting v, denote the present discounted value of cash
flows from the project, and assuming that the interest rate r=0.1, the NPV

of the project is

co fl-K: 1+

(1+ 1’ r’P,-K=$330, (31)
r=1 r

NPV, = v,-K=P,+

 

since the expected price of P from the next period onward is EIP,I =P,
=$80. When waiting is allowed, the solution must again go through the
following two steps: first, we calculate the NPV from the optimal decision in
the next period, when the information on the output price is available.

Next, we compare it with the NPV from the optimal decision in the
current period, and make the final decision from the perspective of the entire
planning horizon. In effect, this is the solution technique of a two-period
dynamic programming problem. Algebraically, the NPV is obtained by

solving

1
+r

 

NPv,=Max{lv,-K, 7 5,1511}. (32)

where EOIFJ is the expected net payoff from the optimal second-period

decision. Evaluating the second term of (32) gives

 

‘2 This assumption is needed to keep the firm's risk preference from affecting the

31

 

1 °° P (1+r/K] (P j
“—5 F = ” — = —"—K =$350. (33)
1+r 0/ ') q(;(1+r)‘ (1+r) q r

SinceVo-K= $330 from (31). solving equation (32) yields
NPV,=Max[330. 350]=$350.

As before, the difference between NPV, and NPV, is a value-added by
waiting,
V,=NPV,-NPV,= $20, (34)
When waiting is allowed, we can obtain the same solution using the

option valuation technique. To show this, let V: be the NPV of the

investment opportunity, and V; and V}, be those binomial values of the
investment opportunity valued in each state from the optimal decision in the

next period. Depending on the movement of the price in the next period, V;

and V), are expressed as

 

V; =Max(-1+—r-Pu -I7+r)K, 0) = $770 (35)
r
V; =Max(1:r Pd -/1+r)K, 0) = $0. (36)

Now that we have contingency values of the project in the next period,
we can construct a risk-hedging portfolio by holding a long position for one
unit of the option to invest, and holding a short position for N units of the
output. We now need to impose two requirements in equilibrium to ensure

that there are no arbitrage opportunities: The first and familiar one is that

 

option value. See footnote 8.

32

the portfolio must yield the riskless rate of return in any state. The second
and new one is that we ought to pay a compensation to other investors to
induce them to hold a long position for the output.13 The compensation
should be made at the riskless rate of return, because we assume that
systematic risk is diversified away, and the expected price of output equals
the current price that is constant (EIP,) =Po). Mathematically, this arbitrage
condition is written as

v;,, -NP,-rNPo=(1+r)(Vf-NP0). (37)
and

Vd’d-NP,-rNP0=(1+r)(Vf-NP0). (38)

The term rNPo is the compensation made to other investors. Letting P,=uP,

and P,=dPo,, and solving these equation for V: gives

a: 1-d
" (1+r/(u-dl

 

[’t’up, —(1+rlK) =s350. (39)

The option to invest has the same value that we have derived in (33).

In our example. the dynamic programming and the option valuation
techniques produce the same numerical result. In effect, they are equivalent
in valuing any options on investment. even though they are based on
different solution concepts. Both of the solutions suggest that waiting and
deciding to invest in the next period has a greater value than immediate
investing. We can also interpret our result from a different perspective. The

NPV from the myopic decision is $330, while the NPV from the flexible

 

‘3 In the last example, we do not need this requirement, since the underlying asset is
the financial asset traded in the market, so the compensation is already reflected in the

33

decision is $350. Hence. if the firm makes the myopic decision, it will lose
$350. which is the value of the option to invest in the next available period.
This lost value should be added to the opportunity cost of immediate

investing. The total cost of investing today, therefore, is
K+ NPV,= $500 + $350 = $850.
which is greater than the gross benefit of investing today,
V,=K+NPV,= $500 + $330 = $830.

The explicit consideration of the option suggests that investing in the current
period should be rejected.

The foregoing discussion implies that the firm has three possibilities in
its investment decisions; it will never invest. or wait until the next period, or
invest now. depending on the current output price, P,. The critical values of
P, are determined by investigating whether NPV,$0 and NPV,SNPV,.“
Imposing these conditions on equations (31) and (39), and solving for P,
gives two critical values of P,, which divide the optimal investment rule into
three phases: if P,S$33.3, it is never optimal to invest in either period since
the project has a negative NPV; if P,>$47, it is optimal invest now since
waiting is costly; and if P, is in-between, it is optimal to wait since waiting
is valuable.

We can also do some comparative statics to investigate how each

variable affects the value of the option to invest. The most interesting

 

asset price in equilibrium.
‘4 The equality in NPvdsNPV, is called the ‘value matching condition' , which is used to
obtain a critical value of Pa. When the time is continuous, an additional condition,

34

variable would be the uncertainty over the output price. We can use (u-dl as
a measure of uncertainty, because a greater value of this is translated into a

more volatile movement of the price in the next period. Partially

differentiating V: with respect to (u-d) gives

(+) (+) (+)

6V" 1+r 1+r
BTU—5717 =(u-7l (TuP, - (1+r/K) +(u-dll7-dl-7—UP0 >0:

which shows that V," is positively related with uncertainty. This results

from the fact that the firm has the flexibility of admitting upside gains. while
dismissing downside losses. by obtaining information about price in the next
period. Waiting may be the firm’s optimal decision in response to
irreversibility of its investment. This also implies that behind the option
value lie asymmetries in the firm’s behavior; see Dixit and Pindyck (1994),
Chap. 2, for other interesting comparative static results.

The basic message of our discussion is that a positive net present value
of a project does not necessarily warrant the acceptance of an investment.
When the investment decision is irreversible, it eventually involves the
option to wait and decide in the next available period. This flexibility has a
value that is to be lost. when the firm decides to invest now. The option
value must, therefore, be added to the cost of an immediate investment.

One qualification is in order; options may work in a perverse way. Waiting

 

called the “smooth pasting condition”, is required at that critical value to ensure a
continuous differentlability. We will discuss these conditions again later on.

35

may induce a substantial cost to firms. For example, by delaying an
investment, the existing firm may encourage other competing firms to
undertake a preemptive investment, so the firm's growth options and value
would be seriously affected by this possibility. 0n the other hand, an early
investment may reveal valuable information that is crucial to a firm’s future
profitability. Marketing, advertising, and R&D expenditures are examples. In
whatever characteristics options may have. the naive net present value
method does not allow for these possibilities. Our simple illustration
provides a persuasive case for casting a skeptical view of the standard
approach to optimal investment decisions. By employing such elegant
techniques as dynamic programming or contingency claim analysis, the new
approach would offer a better alternative in capturing options and providing

useful interpretations.

 

Chapter 2

Uncertainty, Irreversibility, and Tax Policy

2.1. Introduction

Theoretical developments to study the firm’s investment decision have
been enriched by recent research programs, stressing the firm’s uncertain
environment and rigidities in adjusting the capital stock. New theories have
provided valuable insights into the conventional cost-of—capital approach,
and the q-theory of investment: the firm’s cost of capital, and q are no
longer static. The literature on irreversible investment has found that the
firm’s investment process is characterized by the option value, hysteresis
(inertia), and dynamics of rational expectations. These studies have also
offered good empirical models to better explain the observable behavior of
the firms' investment in a more realistic setting.‘

Incorporating taxes into these investment models with uncertainty and
rigidity appears to be a promising research agenda. Nevertheless, few
attempts have been made so far. Along the line of the standard neoclassical
theory, Bernheim and Shoven (1989) employ a capital asset pricing model
(CAPM) to address the importance of uncertainty and risk. They find that
risk premia are a very important component of the cost of capital, and that

the U.S. corporate tax system discriminates against risky investment

 

‘ There is a large amount of literature devoted to these research efforts. See the
survey articles by Dixit (1992) and Pindyck (1991). Dixit and Pindyck (1994) is
their co-authored book, which provides a systematic treatment of the new approach
to the firm's capital decisions.

36

37

projects. In order to treat taxes, risk, and interest rates simultaneously,
they use a highly stylized neoclassical investment problem that permits firms
to adjust the capital stock instantaneously.

MacKie-Mason (1989) studies an interaction between non-linearities in
the tax system and uncertainty in a framework of irreversibility investment.
He employs a stochastic equilibrium model popularized in financial
economics. It is a well-established result that the value of a future
investment opportunity under uncertainty is significant, so that it is optimal
for firms to delay the investment. unless immediate profitability from the
investment is sufficiently high to compensate for this lost value of the option
to wait; see, for example. McDonald and Siegel (1986). Their introduction
of non-linearities in taxes yields a surprising result that the government’s
role of risk-sharing by taxing firms may encourage investment.2

While the work of MacKie-Mason is an application of the Black-Scholes
model of financial options, McKenzie (1994), and Hassett and Metcalf
(1994) adopt a dynamic programming approach to irreversible investment to
consider the effects of taxes.3 McKenzie calculates the marginal effective
tax rates for major Canadian industries, using a model of irreversible
investment. The rates are found to be increasing in both market and specific

risk. He concludes that the tax system in Canada is distorted against risky

 

2 This argument is analogous to one propoSed by Gordon (1985). Yet they differ in
the nature of the risk addressed. Risk is commonly classified into two types: firm-
specific (unsystematic) and aggregate (market or systematic). Gordon's risk
sharing is concerned with market risk only, while Mackie-Mason considers both
market and specific risk.

3 Bertola (1988) shows that two approaches are equivalent in essence, only
differing in the assumption about the tradability of risk in the financial market. See
Dixit and Pindyck (1994), Chap. 4, for a discussion.

38

capital to a greater extent, when irreversibility is explicitly considered. The
Hassett and Metcalf study focuses on uncertainty over tax policy as well as
over the output price. Specifically, they model a random movement of the
investment tax credit to examine the effect of tax policy changes on
investment. Interestingly enough. they argue that different assumptions
about tax uncertainty may have contradicting conclusions: when uncertainty
over tax policy is modeled as a continuous random walk process, which is
popular in most studies. increasing uncertainty discourages investment. 0n
the other hand, when uncertainty is modeled as a jump (Poisson) process, an
increase in uncertainty stimulates investment.

In this chapter, we examine the effectiveness of tax policy in a model
of investment uncertainty and irreversibility. We extend existing studies in
two directions. First, we adopt a more realistic assumption on the firm's
irreversibility in investment. Following Dixit (1989), a firm’s investment is
partially reversible, so that it has the option to abandon the project as well
as the option to invest. Other tax models of irreversible investment
—-including McKenzie (1994) and Hassett and Metcalf (1994). cited
above—usually do not allow for partial reversibility. These models,
therefore, are inadequate in examining the role of irreversibility at various
levels. In the present model, we can carefully identify the effects of tax
policy under different degrees of irreversibility. Also, by taking an eclectic
position between the standard neoclassical model assuming complete
reversibility. and the standard irreversibility investment model assuming a

complete irreversibility, we are in a better position to connect two views.

39

Second. most existing studies focus on deriving the optimal investment rule
by solving a firm’s optimization problem. We will go beyond that by
simulating the aggregate investment behaviors based on the optimal
investment rule. These simulations are useful in verifying the model's
validity in predicting the actual behavior of investment. In doing so, we
introduce a simple heterogeneity in the degree of irreversibility of firms;
firms are differ in their capabilities in reversing their investments. Modeling
the heterogeneous characters across firms has increasingly been employed
by investment theorists. This strategy allows us to point to the fact that
only a fraction of them alter their investments in response to a shock over
time.‘

We begin by providing in Section 2 a little essential mathematical
background for stochastic processes, and stochastic calculus. Section 3
presents a simple dynamic model of a firm's investment decisions under
uncertainty and irreversibility. We solve the model by dynamic programming
techniques. and interpret the resulting solutions for the value of the firm in
terms of the options. We also derive a refined version of the cost-of-capital
formula, and discuss its relationship to the Jorgensonian formula. In section
4, we conduct a sensitivity analysis to examine the effects of taxes.
uncertainty. and irreversibility on the cost of capital. Since closed-form
solutions are not available for the model, we present numerical results

I

instead. In section 5, we introduce a structure of heterogeneity in the firms

 

‘ Some recent models employ a more elegant method of modeling the firm's
heterogeneity by using, for example, an adjustment hazard function. The partial
adjustment model, or the standard (S, s) adjustment model is a special form of the
hazard function. See Caballero and Engel (1992) for these modeling efforts.

40
irreversibility of investment, and perform simulations to generate the
aggregate investment behavior. The simulation results are presented for both
costless and costly reversibility. and for varying parameter values for the tax
system, uncertainty, and irreversibility, to facilitate comparisons. Section 6

is concluding remarks.

2.2. Mathematical Preliminaries

2.2.1. Stochastic Processes and Brownian Motion

A stochastic process deals with a system which evolves according to a
specified probabilistic law. To model uncertainty properly, we must assume
a specific random process which underlies the variables in question. One
example is a discrete-time random walk with drift which evolves according

to
x,-x,-, =p+s,, (1)

where p represents the trend, and a, a white noise, or a random variable
drawn from the normal distribution with mean zero and unit variance. In
the limit as the time interval goes to zero, the discrete-time random walk

takes a process of the (absolute) Brownian motion with drift,
dx=,udt+a'dw, ' (2)

where dw=s,, Els,)=0, and Var(3,) = 1. w is the standardized Wiener
process whose increment dw has mean zero and variance dt. Similar to its
discrete equivalent, p represents the trend, and a the volatility of the

process. Note that x has a normal distribution. so EIdx)=pdt. and

41

Varldx)=a’dt. Over an infinitesimal time interval dt, the Brownian motion
evolves with mean ,udt and variance azdt.
Since this Brownian motion measures the changes in x in absolute

terms, x can take a negative value. If we need to restrict x to positive
. . dx

values, we can assume Instead that the relatIve change of x, —, has a
x

normal distribution to prevent x from being negative.5 Thus we have the

geometric Brownian motion with drift,

2’: =,udt+a’dw. (3)
x

This stochastic process is widely used in the empirical analysis of asset
prices and exchange rates. etc. Ignoring the volatility term. we are left with

an ordinary differential equation,

dx
— = X, (4)
dt [1
which has the solution.
X =Xoeflt, (5)

where x, is the initial value of x at t=t,,. Thus, the geometric Brownian
motion combines continuous-time dynamics with uncertainty in a
mathematically tractable way. It is useful in addressing many typical

stochastic settings in economics and finance.

 

5 This modification is also necessary because, if the absolute change. dx. is
normally distributed, a variation of 1 percent is more significant at low base than at
high base. Hence we need to weight dx by the level value. x.

42

2.2.2. Ito's Lemma

When we differentiate the functions of a Brownian motion, ordinary
calculus is not readily applicable due to the existence of the volatility term.
Consider a stochastic function, y=f(x), where x is a random variable that
follows a process of the absolute Brownian motion of equation (2). When
there is no volatility term, the function fix) is deterministic, so we can apply

Taylor’s theorem to obtain

dy = f’lxldx + gf'lxlldxlz +
= f ’(xlpdt+ oldt). (6)

where oldt) is the sum of higher-order terms in dt that will disappear, when

dt goes to zero. This is the basis of ordinary differentiation.

When we reintroduce the volatility term, -21-f'(xlldx12. the second term in

the first equation above, has a term that involves a square of dw=s,\(dt .
Since the order of this term is dt, we must consider this term in

differentiation. Expanding flx} using the Taylor series expansion yields
dy = f’lxldx + %f'lxl rdx)’ +

= f’IxH/zdt + qdw) + %f’(xl(pdt + a'dwl’ + (7)

Using the fact that Eldwl=0, and Varldw) =Eildwlzl =dt. and SUDPFBSSINQ

the higher terms in dt. the stochastic function dy has mean

Eldy) = [f’lxlp + é-f'lxlazldt.

43

and variance
Varldyl = [f’fxl alzdt.

Recalling that E(dx) =pdt and Varldx) =q’dt in the absolute Brownian motion,

dx=pdt+adw, y follows another process of Brownian motion,

dy = [f’(x)p+ éf”lx}02]dt+ f’lxladw. (8)

When x follows the geometric Brownian motion of equation (3), we can

derive, by taking similar steps.

dy= [f’Ixpr-I- -;—f”(xlx20"]dt+f'lxlxa'dw. (9)

This is a basic result of ltO’s Lemma. In a nutshell, Ito's Lemma is to a
function of random variables what Taylor's theorem is to a function of
deterministic variables. Hence the lemma is used to differentiate a function
of a random variable with respect to the random variable itself.

In ltO's Lemma, the second moment of a distribution plays an important

role. The sign of this term, %f”lxlaz, is dependent on the curvature of the

function, fix). It is the consequence of Jensen's inequality, which relates
the expected value of a function of a random variable, Elflxll, with the value
of the function of the expected value of the random variable, flEfxll; if the
function is strictly convex (concave). the former (latter) is greater than the

latter (former), so that

[fix/I) flElxll. (10)

44

A good example of a convexity case is the profit function. Let fix) be a
profit function. and x be the output price. It is well-known from the
microeconomic principle that fix) is convex in x. Suppose that a firm faces
x that is fluctuating randomly, as in the case of the Brownian motion
process. Then, the inequality relationship (10) implies that the firm can
expect more profits by producing a flexible output in response to fluctuating
prices, rather than by producing a fixed output at the average price. In this
simple example, flexibility has a value; see Varian (1992), Chap. 3, for more

discussion on this.

2.2.3. Application of Ité’s Lemma

Since ItO’s Lemma is so important in working with any function of the
Brownian motion, we will present two functional forms frequently used in
economics: see Dixit (1993) for more general treatment. The first one is of
the form fix)=expi/ix), where x follows the absolute Brownian motion.

Applying ltO's Lemma of equation (8) gives

df= Iiexprixip + % Azexprzxiaidt + .iexprzxiadw
= fix/(.1). + glzazldt + flxuadw, (1 1 )

which can be rewritten as

f’fI. =12,” éfl’a’]dt+ladw. (12)

45

This is nothing other than a form of the geometric Brownian motion. Since

fixl=expilxl is convex, we have a positive term, £1202.

As an application to the geometric Brownian motion, suppose that
fix)=x‘, where x follows the geometric Brownian motion. Using ltO’s

Lemma of equation (9). we can write df as
df= [Ax-u“ é/lil— rixi-Zx’azidt+2.xi-'xadw, (13)

which can be rewritten as

5:1 = [241+ gill-lmzlduzadw. (14)

Observe that fix) also follows the geometric Brownian motion; see Dixit and

Pindyck (1994), Chap. 3, for more discussion.

2.3. The Basic Model

The model closely follows the work of Avinash Dixit (1989). which
analyzes a firm's entry and exit decisions under uncertainty. A
representative firm is infinitely lived, and has a single stand-alone
monopolized project to invest in.° The firm produces one unit of output at
each unit of time. The firm’s uncertainty is over demand, so that, given the

firm's fixed flow of output, price uncertainty corresponds directly to demand

 

° Another modeling strategy is to consider an incremental investment. This would
allow for a more flexible treatment of a firm’s production and adjustment
technology, but at the sacrifice of simplicity of the model. See Abel and Eberly
(1996) for a model along this line.

46

uncertainty, and is considered to be exogenous. The firm can purchase a unit
of capital at a lump-sum price E, when investing, and can sell it at a lump-
sum price R, when disinvesting. We assume RsE, to exclude a costless
arbitrage opportunity. In other words. part of the capital expenditure is
irreversible and is considered sunk, when R<E, while the capital expenditure
is completely recoverable, when E=R.7 The firm incurs an operating cost C
for production. The values of E, R, and C are all constant over time. The
firm pays a corporate tax at a rate r on the flow of its operating income.’3
Finally. the market price of output P is assumed to follow a geometric

Brownian motion process,
dP=ant+anw, (15)

where a is the drift rate parameter, a- is the proportional variance parameter,
and dw is an increment to the standard Wiener process. This equation says
that the return on the project. dP/P, evolves with mean adt and variance
' a’dt over an infinitesimal time interval dt.

Since we consider a discrete investment. the firm's problem is to make
a binary investment decision: it must decide at each instant of time whether
to invest (enter) or disinvest (exit). Let V,iP) denote the value of the

investing firm, and V,iP) denote the value of the disinvesting firm. Assume

 

7 The Dixit model imposes an exit cost to impose irreversibility in investment. Here,
irreversibility is introduced by the discrepancy between the purchasing price and
the selling price of capital. This difference in assumption does not affect the
results of the model. For possible causes for irreversibility, see footnote 1 in
Chapter 1.

' A simple treatment of taxes would suffice for our purpose of comparing the
effects of tax policy with those of other parameters. Allowing for more tax
components is straightforward, as shown by McKenzie (1994) in a model of
incremental investment.

47

that the firm is risk-neutral, and maximizes the expected value of cash flows
discounted at the after-tax risk-free interest rate. V,iP) is then the expected

present value of operating net cash flows from investing,
v,rp,) = mix 5, (:21 - mp, - C)e‘""’"ds .9 (16)

This maximization problem can be solved either by the dynamic
programming method or by the contingent claims analysis; see Chapter 1 for
a discussion of the solution techniques. As we have seen in a discrete-time
framework, the dynamic programming technique is very useful for solving
sequential decisions, especially when time and uncertainty are combined.
Now we must work with a model in a continuous-time framework. The basic
solution technique is to apply Bellman’s Principle of Optimality: an optimal
solution has the property that. whatever the previous decisions, the
remaining decisions must be optimal given the current state of nature. Its
mathematical form is known as the Bellman equation; see Dixit and Pindyck
(1994), Chap. 4, for a good exposition of the Bellman equation. Since we
assume that the firm has an infinite planning horizon, the optimization

problem can be expressed as a recursive Bellman equation,
V,iP) = ii-tliP-Cldt + e””"""E[V,iP + dP)]. (1 7)

In the Bellman equation above, we split the integral into two components:
the first term is the immediate cash flow from time 0 to dt. and the second

term is the expected value of the firm from dt to infinity. Note that

 

’ We assume implicitly that interest payments are deductible. Then, the opportunity
cost to the firm of investing in risk-free nominal cash flows is the after-tax interest
rate, when the firm is free to borrow or lend money.

 

48

e””""" = i-rii-rldt + oidt).
and

E id V,) = El V,iP + dPll- V,iP).
oidt) is the sum of higher-order terms in dt that can be ignored, when dt
goes to zero. Using these facts, and ignoring higher-order terms in dt, we

can simplify equation (17) as

V,iP) = ii-t) iP-Cldt + [1-ri1-tldt]{V,-IPI+ EIVJP + dP/I- V,iP)}

= i 1 - r) iP-Cldt + V,iP)-ri 7-1) V,iP)dt + E[V,iP + dPII- V,iP)}.
Therefore, we obtain
ii-t)rV,iP)dt = i1-r)iP-C)dt+ EidVJ. (18)

Equation (18) is an arbitrage condition. The left-hand side is the after-tax
required return on owning the firm. The right-hand side is the expected
return consisting of the instantaneous after-tax profit flow plus the expected
change in the value of the firm. This is the fundamental asset equilibrium
condition.

Since V, is a function of a stochastic variable, P, that follows the
geometric Brownian motion, we cannot use the ordinary derivative to expand
EidVJ. Instead, we must invoke ltO's Lemma from the previous section to

get

EidVJ = aPV,’iP)dt + £01132 V,'iP)dt. (1 9)

Recall that the term, gazP’V/iPldt, is from Jensen's inequality, so it is

called the Jensen-lt6 effect. When the function V,iP) is of a strictly convex

49

or concave form, the effect activates. Note that the Jensen-Ito effect has
nothing to do with risk attitude of the firm, since we are maximizing the
expected value of cash flow, not the expected utility of the cash flow.
Substituting equation (19) for EidVJ into equation (18). and dividing by dt

gives a second-order differential equation,
éa’Pz V,-"iP) + aPV,’iP)-i1-r)rV,-iP) + i1-rliP-C) = 0. (20)

This is the equation of motion for V,-. which characterizes the evolution of
the value of the firm, V,. The condition for optimal investment is derived by
solving the equation.

When the cash flows from the project have a market risk, and the firm
behaves as if it is averse to risk, we ought to take risk aversion into account
in valuing the project. Two treatments are available. One method is to use
the risk-adjusted discount rates to discount cash flows. In our model,
however, finding the appropriate risk-adjusted discount rate is impossible,
because the risk of the project changes every time the output price
fluctuates; see the relevant discussion in Chapter 1.

The alternative is the certainty-equivalent method, by which we
calculate the certainty equivalent of risky cash flows, and discount it with
the risk-free interest rate. In the stochastic process assumed in equation
(15). the comparable task is to replace the actual drift parameter a, with the
certainty equivalent drift parameter. To do this, let r, be the risk-adjusted
market equilibrium rate of return from holding the project. Then, from the

fundamental condition of equilibrium for the capital asset pricing model

50

(CAPM), the equilibrium rate must be greater than the risk-free interest rate

by the amount of risk premium, such that
fl: =r+ ¢ppmal (21)

where r is the risk-free rate of return, (6 is the market price of risk, ppm is the
correlation of the return on the project with the return on a market portfolio,
and a- is the standard deviation of the project’s return in equation (15).‘°"1
As we have mentioned above, r, cannot be used directly to discount

cash flows. Instead, we explore the relationship between r. and a to obtain
the certainty equivalent of a. Since r. is the equilibrium discount rate and a
is the rate of price appreciation, r, should be greater than a in market
equilibrium. If this were not the case, the firm would never invest, because

waiting always pays. Therefore, we have a relationship,

fk=a+§, (22)

where 6, the difference between r. and a, is always positive, and reflects an

opportunity cost of delaying the project. Equating (21) and (22) gives

ot-¢pp,,,a'= r-5. (23)

 

‘° Notice that p,,,,0' =Covlp,m)/cr,,,, where CovIp,m) is the covariance between the
project's return and the market return, and am is the standard deviation of the
market return. It stands for the market risk of the project. By a proper

transformation, this term can be expressed as p,,,,cr= ,3- a',,,, where 5 =Cov(p,m)/0’:

is the market beta of the CAPM. The market beta measures how the project's
return is sensitive to the movements of the market return.

" To invoke this relationship, we must assume that the project is directly tradable
in the asset market. or that we can construct dynamic portfolio of assets whose
expected return is perfectly correlated with that of the project. This is an
underlying principle of contingent claims analysis, or financial options theory. See
Varian (1987) for heuristic approach. See Dixit and Pindyck (1994), Ch. 4, for an
application to investment analysis.

 

51

This equation says that a-¢p,,,,q, the certainty equivalent of the drift rate a.
equals r—J in equilibrium.12 Now, replacing a by its certainty equivalent

value, r-5, in equation (20) and adjusting this by the tax rate. r, we get
gen-2P2 V,'iP) + i1-r)ir-6‘)PV,’iP)-ri 7 -r) V,iP) + i 1-r) iP-C) = 0. (24)

With this equation of motion with risk aversion, we can now continue to
solve for the value of the investing firm, V,iP).

The solution for V,iP) consists of two parts: the complementary function
(the general solution of the homogenous form) and the particular integral
(any particular solution of the non-homogeneous form). Note that this

differential equation is of the Cauchy-Euler type, so we can take a trial

solution P” to get the complementary function. Substituting this for V,iP)

into the homogenous form of equation (24) gives
Pl % a’flifl- 1) + (1-r)ir-6)fl-ri1-r)l = o. (25)

Solving this equation for B, we get13

 

I31=-;--i1-r)ir-6)/o’+ Jm- tlir -6l/a’2 42-12 +2ri1— zI/a’ >1

 

I32 =é-i1-tlir-5l/az-Jli1- rlir — a) / 0'2 - £12 + 2m - r) / a" < o.

 

‘2 Notice that this method is closely related with the risk-neutral probability method
discussed in section 1.2 of Chapter 1. It was first pr0posed by Constantinides
(1978). Its theoretical results are based on the intertemporal CAPM analysis
pioneered by Merton (1973). Brealey and Myers (1991), Chap. 9, provides a simple
exposition of how to use the CAPM to calculate a certainty equivalent.

‘3 Denoting the expression in bracket in equation (25) by Difl), we can see that

0(0) =-rii-r)<0, and 0(1) =-6i1-r)<0. This implies that one root is greater than 1,
and the other is less than 0.

52

Since the two roots are real and distinct, the complementary function is
V,‘iP)=A,P”' +A2P’2, (26)

where A, and A, remain to be determined.

The particular integral can be obtained by the method of undetermined
coefficients. The obvious form of the solution would be V,iP)=B,P-B,C, so
that V,’iP)=B, and V,"(P)=0. Substituting these into equation (24) and

collecting terms, we get
JB,P-rB,C = P-C. (27)

Equating both sides term by term gives 8, = 7/5 and 8,: i/r. Thus, the

particular integral is
V,” (P) = P/6- C/r. (28)

Finally, we can combine the complementary function and the particular

integral to obtain the complete solution,
V,iP) =A,P’i #1,)”32 +P/6-C/r. (29)

This equation has a straightforward economic interpretation. The last
two terms of the right-hand side are the particular integral. They stand for
the fundamental value of the (investing) firm that stems from the expected
net cash flows. The first two terms are the complementary function. They
stand for the value of the option to exit that must be added to the

fundamental value of the firm. If the investment is completely irreversible,

53
the option to exit will disappear, leaving only the fundamental value of the
firm. This option value becomes worthless at a higher level of P, since it is
less likely that P will drop to the critical level, P,, that induces the firm to
exit. At an extreme, where P goes to co, the value of the option goes to
zero. Yet the first term of (29) goes to 00 as P goes to 00, because fl,> 1.
This implies that A, should be zero. Thus, the imposition of a limiting

condition at P=oo yields the value of the investing firm,

V,iP) =A2Pfiz +P/6—C/r, for P>PU. (so)

The value of a disinvesting firm, V,iP). can be derived in a similar
manner. We can think of it as the mirror image of the investing firm; if the
firm exits, it gains the option to invest, while losing the operating profits.

Its value is the option to invest only. Similarly to equation (17), the value of

the disinvesting firm is written as
V,iP) = e"”"""E[V,iP+ dP)]. (31 )

Taking similar steps as before, we have a differential equation.
.2. 02P2V,'(P) + i1-r)ir-6)PV,’iPl-ri1-tl V,iP) = o. (31)

Note that this equation has only the homogenous part. Solving for V, gives
V,iP) =B,P”' +32)”. (32)

The two terms on the right-hand side stand for the option to invest. This
option value becomes worthless at a lower level of P, since it is less likely
that P will rise to the critical level, P,, that induces the firm to invest. At a

extreme, where P goes to zero. the value of the option goes to zero. Yet the

54

second term of (32) goes to 00 as P goes to 0, because ,6,<0. This implies
that B, should be zero. We are now left with the value of the disinvesting

firm,

V,iP)=B,Pfi', for P<P,. (33)

We have to solve equations (30) and (33) for A2, 8,, P,, and P,. To do
so. we need two boundary conditions at the threshold prices, P, and P,.
Recalling that E denotes the purchasing price of capital, and R denotes the
selling price of capital, let us first examine how a firm behaves at the
threshold prices. We can distinguish the firm's current operating status by
the option it has: an inactive status that has the option to invest, and an
active status that has the option to disinvest. The inactive firm will decide
to invest. and will earn V,iP)-E in exchange for V,iP). when P rises to P,.
The active firm, on the other hand, will decide to disinvest. and will earn
V,iP)-I-R in exchange for V,iP). when P falls to FL. The firm will alternate
between the decision to invest and the decision to disinvest. depending on
the price level that fluctuates over time. This implies that V,iP)-E= V,iP) at
P,, and V,iP)+R= V,iP) at P,. This is called the value-matching condition.
The other condition, called the smooth-pasting condition, is directly related
with the value-matching condition; it requires that derivatives of each set of
the value functions above should also have the same value. when evaluated
at P, and P,. The smooth-pasting condition ensures a smooth transition in

going from one regime to the other.“

 

“ A rigorous treatment of these two conditions requires stochastic optimal control
theory. For more details. still at an introductory and intuitive level, See Dixit
(1991a.1993L

55

Imposing these two conditions at P, and P, gives four equality

conditions,

Var/Pu) = V,iPUI-E, V,iPu) = V,-'iPu) (34)
and

WP.) = Var/PL} + R. v,’rP,) = v,'(P,). (35)

Substituting equations (30) and (33) into equations (34) and (35) for V,iP)
and V,iP). respectively, we have a simultaneous equation system, consisting
of four equations and four unknowns: A2, 8,, P,, and P,. These equations
are nonlinear in most of the parameters, so we need numerical solution, to

which we proceed in the next section.

2.4. Tax-Adjusted Cost of Capital

The original Dixit model proposes two important results that qualify the
standard neoclassical analysis. First. the optimal behavior of the firm
comprises three regimes: (i) the firm will invest, if P is greater than Pu; (ii)
the firm will neither invest nor disinvest. if P is between P, and P,; and (iii)
the firm will disinvest, if P is less than P,. Second, a region of inaction
between P, and P, is wider than the conventional Marshallian one. The
introduction of taxes in the present model poses the following questions: to
what extent do taxes affect these two critical prices, and how do taxes

interact with uncertainty and irreversibility in effecting them?

 

56

To build a base in answering these questions. let us rewrite below the

six key equations which characterize the solution to the firm’s investment

problem,
-21-a'2P2V,-"iP)+ i1-r)ir-6)PV,’(P)-ri1-r)V,iP)+ i1-r)iP-C)= o. (36)
é-q’P’ V,"iP) + i1-r)!r-6)PV,’iPl-ri1-t) V,iP) = o; (37)
V,iP) =A2Pfl" +P/5-C/r, for P>P,,, (38)
V,iP) =3, P“, for P<P,; (39)

and

Valpu/ = Vilpul’E, Vd'lpu) = Vi'lpu), (40)
V;(PLI = lepl.) +R, Vi'IPL) = Vd'lplj' (41)

Equations (36) and (37) are the equations of motion for V,iPland V,iP).
respectively.

By solving these differential equations separately. we get the value of
the active firm and the inactive firm expressed in equations (38) and (39).
Equations (40) and (41) are the value-matching and the smooth-pasting
conditions at each threshold price. Inspection of equations (38) and (39)
shows that the tax does not affect the fundamental value of the firm, P/6-

C/r, since it only affects the value of the firm through its effect on the value

of the options, A, Pfi’ and 8,1)“ . To investigate the relationship between

equations (38) and (39) further, define WiP) to be

WiP) = V,iP)-V,iP) =P/5-C/r—B,P” +A2P‘2. (42)

 

57

WiP) is the changes in the value of the firm that shifts from an inactive
status to an active status. Observe that WiP) has a direct bearing on
Tobin's q. Tobin's q is defined as the ratio of the market value of installed
capital to its replacement cost. In our specification, WiP) is the numerator
of Tobin’s (marginal) 0, that is, the market value of installed capital. It is
the sum of three components: (i) P/rS-C/r is the present value of expected
profit flows, if the firm has neither the option to invest nor the option to

disinvest; (ii) If the firm has the option to invest, this opportunity reduces
WiP) by B,Pp‘ , since the option disappears by investing; and (iii) If the firm

has the option to disinvest. this opportunity increases WiP) by AzP”, since
the option arises from investing. This is a modified version of Tobin's 0.
when the firm's options on investment are explicitly considered.

We can also relate the equations of motion (36) and (37) to the cost of

capital. Subtracting equation (37) from equation (36) gives
éa’P’W'iP) + ii-rlir-cSIPWiPI-rii-r) WiP) + ii-rliP-C) = o. (43)

Rearranging this yields

1
2i1-rl

 

P = c + rWiPI-[ir-6IPW’iP) + o’P’ W'iPII. (44)

The left-hand side, P, is the marginal revenue product, since we assume a
unit production. The right-hand side is the cost of capital, consisting of (i)
operating cost. C. Iii) interest cost on the market value (shadow price) of

capital, rWiP), and (iii) the expected value of the capital loss (gains).

 

58

7
2(7—7)

 

ir-6)PW’iP)+ o’P’W'iP). Thus, equation (44) stipulates the firm’s

optimal investment condition that the marginal revenue product is always
equal to the marginal cost of capital.

It is useful to evaluate (40) and (41) at each threshold of P to get

WlPu) = V;(P,l-V,(Pul =5, and Wllpu) = V,iPuI-VJIPUI = O, (45)

WIP,) = V,iP,)- V,iP,) = R, and W'iP,) = V,'(P,)- V,'IP,) = O. (46)

These relationships state that the market value of capital, WiP), equals the
purchasing price of capital, E, or the selling price of capital. R, at each
threshold price, and that its derivative evaluated at each threshold price is

zero. Using these relationships, we can evaluate equation (44) at P=P, to

 

 

get

P, = c +rE- 2 11- “ a’Pj W'iP,) > c +rE (since W'iP,) < 0). (47)
Similarly at P,, we have,

P, = c + rR- 2 1: I) a’Pf W'iP,) < c + rR (since W'iP,) > 0). (48)

Equations (47) and (48) imply that
P,<C+rR< C+rE<P,.
C+rR and C+rE are the Marshallian threshold prices that offer the

benchmarks of the optimal investment behavior in the neoclassical

framework. P, and P, deviate from them due to the term, 3711—502P’W'iP).

We now investigate how P, and P, behave in response to changes in the tax

59

rate (r). uncertainty (0'). and irreversibility (R) by numerically solving the four
equations in (40) and (41). The base parameter values are set to: r=0.04,
6=0.04, a=0.2, r=0.3, C=0.8, E=2, and R=2.‘5 Consider now three

cases that are illuminating.

Case 1: E=R=2.0; ajnd tvary.

 

This is the special case of costless reversibility, on which standard
neoclassical investment theory is based. Recall from equation (23) that the
certainty equivalent of the drift parameter, a, is expressed as a-lp,,,,a-=r-6.
Hence, by setting r=6=0.04, we are assuming the zero risk-adjusted drift
term. so that we can focus only on the effect of the volatility term on the
option values. In this special setting, the values of the two options in
equation (42) exactly offset each other, and we are left only with the
fundamental value of the firm. Figure 2.1 and Table 2.1 shows the effects
on the threshold prices, P, and P,, of a combination of parameter values for
a- and r. Observe that P, and P, collapse to the conventional Marshallian

threshold, PM = C + r5 = C +rR = 0.88, regardless of the values of a and r.

 

‘5 The base parameter values closely follow those used in an example of Dixit and
Pindyck (1994). Chap. 7. Since a—¢p,,,,a=r-6 from (23), we assume by setting both
I and 6 at 0.04 that a must adjust in response to the change in a, so that the two
terms on the left-hand side cancel out each other. This implies that 6 is a
fundamental parameter, while a does the adjusting.

60

Figure (Table) 2.1 - Effect of the Tax and Uncertainty on the Threshold
Prices (Costless Reversibility: E=R=2l

1.2

 

 

 

 

 

 

 

 

 

0. 6
.5 4 .4 -5
° .3 - 3
, 2 . 2
0' . l . 1 T
O
0'
1' 0 0 1 0 2 0.3 0 4 0 5
Pu
O 0.88 ° ' ° ' 0.88
0.1 o o o o e o
0.2 o o o o o e
0.3 o o o e o
0.4 e o o e o o
0.5 0.88 ‘ ' ° ° 0 88
PL
0 0.88 ° ' ' ' 0.88
0.1 e e e e o o
0.2 o o o o o o
0.3 o o o o o o
0.4 e o e o o o
0.5 0.88 ' ° ° ' 0.88

 

61

This is because W'iP) in (47) and (48), evaluated at P, is equal to zero.” In
this special case, both a corporate tax and uncertainty have no effect on the
threshold prices. This result is also consistent with the well-established

proposition that a profit tax is neutral in the standard neoclassical model.

Case 2: E>R; a' and r vary.

This is a more general case of costly reversibility. Figure 2.2 and Table
2.2 illustrate the effects of reducing the selling price of capital from 2.0 to
R: 1.9. We can see that even a small sunk cost has a significant effect on
the cost of capital. The picture also shows that the effect of uncertainty
dominates that of the tax throughout the whole range. At a higher level of
uncertainty, the region of inaction is substantially widened. The base-case
simulation in the table shows that P, increases by 0.04, and P, decreases by
0.03, respectively, as r rises from 0 to 0.5. On the contrary, P, rises by
0.34, and P, falls by 0.23, respectively, in response to the corresponding

change in c.

 

‘° This can be confirmed by the graph below. WiP) passes a point of inflection at
PM=Oo88.

 

 

mp)
10
7.5
5
2.5
p
.25. 0.5 1 1 5 2
-5:
-7,5.
-10.

62

Figure (Table) 2.2 - Effect of Taxes and Uncertainty on the Threshold

1.9)

n:

I

=2

Prices (Costly Reversibility: E

 

 

 

 

 

 

63

= 0.2' n tv r.

In this case, we fix the uncertainty, while varying irreversibility and the
tax rate. Figures (Tables) 2.3 and 2.4 show the results for two values of
uncertainty: 0:0 and a=0.2. Table 2.3 shows that irreversibility only
affect the lower threshold price, P,, even under perfect certainty, implying
that the firm is more reluctant to exit when the investment cost recovery is
smaller. In the base-line simulation, P, falls from 0.88 to 0.84 by reducing R
to 1.0 from 2.0. The tax rate changes, however, have no effect at any level
of irreversibility. Adding uncertainty alters the picture in a significant way,
as we can see from the comparison of Figure 2.3 with Figure 2.4. In Table
2.4, the base-line calculation shows that, compared with the Marshallian
benchmark of 0.88, P, increases by 0.40 to 1.28 and P, reduces by 0.28 to
0.60, respectively as E reduces to 1.0 from 2.0 The tax rate changes do
have an effect under uncertainty, but their magnitudes are moderate; the
resulting changes in the threshold prices are less than 0.1 in most ranges of
R. The striking feature of Figure 2.4 is that even a small change in
irreversibility has a strong effect on the firm’s cost of capital, especially at a
lower level of irreversibility.

This observation is reminiscent of a famous argument found in the
menu cost literature. Menu costs refer to the costs of changing prices. For
example, a firm may need to send out new catalogues to announce the price
changes of its products, which incurs some costs to the firm. Menu cost

models were originally developed to explain price stickiness in the short run.

 

64

Figure (Table) 2.3 - Effect of Taxes and Irreversibility on the Threshold
Prices (Certainty Case: a= 0)

 

 

 

 

 

 

 

 

 

 

 

65

Figure (Table) 2.4 - Effect of Taxes and Irreversibility on the Threshold

Prices (Uncertainty Case: a=0.2)

 

 

.

s
.

  

.

.
A
.

.2

9'

$2.
...

.
o
C

,

.
e
.

2

.
.
.

.
.
.
.

s
s

.

.
..

 

 

 

.
.

3
v.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

66

The basic idea of these models is that prices do not adjust immediately since
there are costs involved in changing prices. Sticky prices, in turn, lead to a
large economic disturbance because small costs of adjustment induce a wide
range of inertia: if it is optimal for firms not to change prices, nominal prices
will not adjust. and monetary policy will affect output.

When a firm's capital decisions are costly to reverse, the optimal rule
for investment, or disinvestment, also involves a range of inaction. The firm
would make an investment decision only when profitability is especially
good, and would reverse its decision only when profitability becomes
especially bad. There is an intermediate range where a status quo is
optimal.17

Finally, it would be an interesting experiment to calculate the tax
equivalent of an increase in uncertainty, since 1 and c act together to
change the threshold prices through their effects on ,3. Specifically, we set
R=1.9, and r=0.3, and calculate the threshold prices by increasing a by
0.05 at a time from each benchmark value of 0.2 to 0.5. Then we work
backwards to calculate the tax rate required to obtain the same threshold
prices, with a being set to benchmark values. Table 2.5 shows the result.
Let us take a figure, say, 38.9 from the first column, to give an
interpretation. This number means that we need a 38.9 percent increase in

the tax rate to get the same effect that we would get from raising a- from

 

‘7 The early menu-cost models are static. In these models, a second-order menu
cost causes first-order effects. See, for example, Mankiw (1985). Using a dynamic
menu cost model, Dixit (1991b) shows rigorously that the fourth-order small cost of
change generates first-order inertia.

67

Table 2.5 - Tax Equivalent of an Increase in Uncertainty
(percent. R=1.9)

 

benchmark a

 

 

increment in a 0.2 0.3 0.4 0.5
0.05 25.2 18.6 14.7 12.1
0.1 38.9 30.6 25.2 21.4
0.15 47.1 38.9 33.0 28.6
0.2 52.5 49.2 38.9 34.3
0.25 56.2 44.8 43.5 38.9
0.3 58.8 52.5 47.1 42.7

 

Source: Author's calculations

0.2 to 0.3. If we take a=0.3 as a benchmark, the corresponding increase in
the tax rate required to get the same effect of raising a from 0.3 to 0.4 is
30.6 percent. From this experiment, we can infer that, if the economy is
experiencing a smaller volatility, a stronger tax incentive may be needed to
offset a certain amount of volatility. On the other hand, if the economy is
highly volatile. a less strong tax incentive would be enough to offset the
same amount of volatility.

Two implications follow for tax policy from our analysis. First, the
conditions for the tax neutrality in the standard models need to be
reexamined. It is well documented that expensing of capital expenditure, or
allowing for economic depreciation, will lead to a neutral tax system,
depending on interest deductibility. Yet as we have already seen, a profit

tax is no longer neutral under irreversible investment. If we ignore this

68

possibility, tax policy might be designed in a non-optimal way. It would,
therefore, be a worthwhile effort to reexamine the standard proposition on
neutrality of taxes by allowing for more complexities.18 Second, we have
already shown that even a profit tax may affect the firm's investment
decision. Yet its effect ought to be examined in the context of other
considerations like irreversibility and uncertainty. The driving force of the
present model is the effects on the value of the options of these key
parameters. An option has a greater value. the more flexibly the firm
behaves and the higher the uncertainty is. This implies that asymmetries
(irreversibility) and uncertainty in the economy or in the tax system might
have a stronger effect than might the level of taxes.19 Also, if firms are in
the region of inaction between P, and P,, it is optimal for them to make no
move. The problem with this is that the government is likely to step in and
use tax incentives to stimulate investment, although it is not an Optimal
intervention. The government should be cautious in distinguishing between
these two elusive situations. Even if the case of the optimal intervention is
correctly identified, fine-tuning the investment through tax instruments
would be a tough task, in the sense that the region of inaction is a very
dynamic one, so it is also a region of ignorance. The implication is that it is
almost impossible to conduct tax policy by discretion by taking whatever

policy seems appropriate at each time. The best policy option would be to

 

‘3 Abel (1983a) is an attempt along this line of research in the presence of
adjustment costs in investment. See also Hartman (1978).

" We already mentioned two works along these lines. Hassett and Metcalf (1994)
introduce random taxes into a model similar to ours. MacKie-Mason (1994) studies
an interaction between non-linearities in the tax system and uncertainty, in a
stochastic equilibrium model a la Black and Scholes (1973).

69

announce in advance how policy will respond to various contingencies. This
is why the second-best rule-based tax policy deserves further consideration

under uncertainty and irreversible investment.

2.5. Simulation of Investment Behavior

In the previous section, we have derived a firm's optimal rule of
investment. To reiterate, the optimal behavior of the firm comprises three
regimes: (i) the firm will invest. if P is greater than P,; (ii) the firm will
neither invest nor disinvest. if P is between P, and P,; and (iii) the firm will
disinvest, if P is less than P,. It would be useful and interesting to use our
investment decision model for simulations of aggregate investment behavior.
The first step of doing this work is to generate a sample path of the output
price from the assumed stochastic process. Figure 2.5 is a sample path of
the price over 150 months to be used for simulation. The underlying
stochastic process is a discrete version of the geometric Brownian that

evolves according to

P,=Pr-r +0.0577P2-r 5:!

where P,=0.88, and a, “N(0,1).2°

 

2° A time interval taken is of one month, and the starting value, P,=0.88, is the
standard Marshallian threshold price discussed earlier. This process has only a
volatility term with q=0.2 per year. Its monthly equivalent, therefore, is

02.2 /12 =o.os77.

output price

(14 015 (16 0(7 (18 (19

L2 L3 L4

1A

L0

70

 

WW“
0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

time

Figure 2.5 - A Sample Path of Output Price

71

When there are many firms, each of them will at each point of time
observe the output price and decide whether to invest. stay, or disinvest,
according to its own optimal decision rule. Once investing or disinvesting is
undertaken, firms are free to reverse their actions, but with incurring the
same cost of entry and exit again. Simulations are performed for 300 firms
that are identical except for the selling price of capital, R; we assume that R
is normally distributed across firm, with mean and variance to vary across
simulations. The simple structure of firm heterogeneity will allow us to
highlight the role of irreversibility.

For the purpose of comparison, we first conduct a simulation without
assuming firm heterogeneity, that is. by assuming that investment is
costlessly reversible for every firm. Figure 2.6 shows the result. We can
see a bang-bang pattern in the investment rate, calculated as the number of
currently active firms divided by the total number of firms. As discussed
before, these firms have the single Marshallian threshold price, P=0.88.

In other words. there exists no region of inaction. Thus, when the output
price goes up even slightly above it, all firms will rush into an investment.
On the contrary, when the price goes down even slightly below it, all firms
will rush out of the investment; the investment rate rises immediately to 100
percent in the former case, and falls to zero percent in the latter.

To simulate a case of heterogeneous irreversibility across firm, we
assume that the selling price of capital, R, is normally distributed with mean

1.8 and standard deviation 0.2." The result is exhibited in Figure 2.7.

 

2‘ But, when the randomly generated values were greater than 2, they were
truncated to 2, because R.<E =2 by assumption.

72

L0

 

0.8

0.6

 

investment ratio
0.4

0.2

 

 

 

 

 

 

 

 

 

 

II I 11111 I I 1 lunnnulllnnunn II I l l I I I l I I 9111]

O 10 20 30 40 50 60 70 80 90 100110120130140150

 

0.0

time

Figure 2.6 — A Simulated Path of Investment (Costless Reversibility Case)

73

L0

 

 

05

 

f

Investment ratio
04

02

 

00

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

time

Figure 2.7 - A Simulated Path of Investment (Costly Reversibility Case)

74

When we add a small degree of irreversibility, the rate of investment shows
a remarkably slow and persistent movement. The investment rate rises to a
maximum of 50 percent in the first 60 months, and ends up with 79 percent
at the end of the whole period. The period average is 66 percent. This
picture would be closer to the observed behavior of investment than the
previous one. I

Let us next perform some sensitivity tests, as we did in the last
section, to examine the effects of key parameters. We reset parameter
values to a higher level and run simulations again according to the same
operating rule as before. The three simulation paths are depicted in Figure
2.8. Not surprisingly, all the investment paths exhibit more sluggish
adjustments over time, when compared with Figure 2.7. In the order of
slowing investment, the tax is the least effective, followed by irreversibility
and uncertainty. More specifically, we can compare the mean investment
rate of each path with the benchmark rate of 66 percent in Figure 2.7.

When the tax rate are raised by 50 percent from 0.3 to 0.45, the mean
investment rate falls by mere 4 percentage points to 62 percent. Adding
more uncertainty has a much stronger impact on investment, as can be seen
from the short-dashed line. When uncertainty increases by 50 percent from
0.2 to 0.3, it falls by more than 20 percentage points to 44 percent. The
long-dashed line is the investment behavior for increased irreversibility by
decreasing the mean selling price, R, from 1.8 to 1.7. which implies an
increase of the sunk cost by 50 percent from 0.2 to 0.3. In this case, the

average investment rate falls by 6 percentage points to 60 percent.

75

 

 

 

     
 

 

O
,_' I'
m
0 db
:3
C
L
to b-
..._9 o'
C:
Q)
E ..
a .-
Q)
E — odd tox
._ 9!- --- odd uncertainty
° — - odd irreversibility

 

 

 

 

01)

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150

time

Figure 2.8 - Simulated Paths of Investment (For Higher Parameter Values)

76

Notice that the negative impact of adding irreversibility is salient in the
earlier periods, but once the rate of investment gains momentum, it seldom
goes down over time. This is not the case for the tax and uncertainty. As
can be observed in the solid and short-dashed lines, adding uncertainty or
the tax rate affects investment uniformly throughout the simulation period.
The reason is intuitively clear. When firms are restricted by a sunk cost,
they are reluctant to make a commitment that is irreversible, unless the
prospect for profitability is sufficiently high. But, once they make a
commitment, the perverse behavior works in the reverse way: they are
unlikely to disinvest in order to avoid incurring an entry cost again.
Irreversibility is a primary source of generating inertia in investment.

The simulation results demonstrate how rational expectations work
under uncertainty and irreversible investment. In traditional investment
models, perfect foresight, combined with no adjustment costs of investment,
renders rational expectations almost useless, since the firm's capital stock
can always revert to the optimal path instantaneously and costlessly. This
is as much as to say that firms make investment decisions with static
expectations. Uncertainty and irreversibility in investment bring time- and
state-dependent dynamics back to the firms' capital decisions. In our model,
irreversibility of investment causes the firms to make more intermittent and
non-synchronized adjustments. Rational firms are in full grasp of the
probabilistic law of motion, and wait until the iron gets hot enough. This
cautious, but optimal, decision rule is a source of the sluggishness and

asymmetries observed in the actual investment behavior. Uncertainty,

 

77

irreversibility. and taxes interact to amplify the dynamics of the investment

process.

2.6. Summary and Conclusions

Allowing for irreversibility of investment in the traditional model of
investment generates the value of options. and changes the value of the ,

firm, accordingly. This value is not negligible, causing the cost of capital to

 

deviate from the single threshold that the simple neoclassical theory
proposes. The optimal rule of decision to invest in this setting is a wait-and-
see strategy, rather than an immediate action, leading to the time path of
investment that is slow yet lumpy, and downward rigid. Adding uncertainty
magnifies this effect. The effect of taxes is dominated by the combined
effects of irreversibility and uncertainty. The standard non-constrained
optimality condition fails to hold. Hence we need to reexamine efficiency
and welfare consequences of taxes, including the conditions for neutral
taxes, in a new setting.

The model has several limitations. so we should be cautious in
interpreting the results. Some of the reasons for caution have already been
put forward elsewhere in previous discussions. We assume a simple and
exogenous process for the firm's uncertainty over demand and output price.
Yet the firm's uncertainty is determined by the responses of other firms.
The model can be modified to endogenize the nature of uncertainty by
introducing competition among firms. This extension would be an initial step

toward a more general model of equilibrium to include other important

78

aspects in the firm’s capital decision. The model can be enriched by
introducing additional uncertainty in the firm's capital price and taxes, or by
adding other taxing schemes. We also assume a sunk cost in investing. But
the opposite possibility exits, that is. a sunk cost in not investing. This can
be modeled as costly expandability in investment, together with costly
reversibility.22 For example, the threat of entry by other firms will force
existing firms to speed up investment. This strategic feature would be
important in studying the effects of tax incentives on the firm’s investment
in R&D.

Another promising extension would be to calculate the cost of capital
using the actual data for different types of assets or industries. The most
difficult problem for this line of research is to measure a degree of
irreversibility. A study by McKenzie (1994) cited in the introduction avoided
this problem by assuming complete irreversibility across all industries. This
unrealistic assumption can be relaxed by utilizing some recent attempts to
offer empirical evidence on the costs of reallocation of capital across firms
and sectors. For example, using information on sales prices and the
characteristics of buyers in the aerospace industry, Ramey and Shapiro
(1998) provide empirical results for the costs of reversing investment
decisions. Since equipment might have a greater degree of irreversibility
than structures, this new research attempt might reverse the conventional

view that the U.S. tax system has tended to favor equipment over

 

2’ See Abel and Eberly (1994), and Abel, et al. (1996) for important theoretical
developments in this direction. It seems that much remains to be done in both
theoretical and empirical analysis, however.

79

structures, especially from 1981 to 1986 (See Gordon, Hines, and Summers

(1987) for an empirical study that supports the common view.).

Chapter 3

Tax Asymmetries, Irreversibility, and the Optimal
Investment Rule

3.1 . Introduction

Typical economic models are concerned with linear relationships among
variables. Many tax models are no exception. They analyze linear forms of
taxation, especially in the goods market. A lot of literature on optimal
commodity taxation assumes an ad valorem tax, whose rate is given as a
proportion of a commodity price (e.g., Sandmo (1976)). Many studies on
income taxation also employ a linear income tax schedule, where the
marginal tax rate is constant (e.g., Stern (1976)). In many instances,
however, we should take into account the fact that tax systems are
characterized by non-Iinearities, including kinks, asymmetries, and jumps.

In the investment theory under uncertainty and irreversibility, the (call)
option to investis a convex function of the output price. This option effect
emerges from the fact that firms can capture upside gains, while they can
avoid downside losses. This asymmetry, in turn, implies that the net value
of the firm may be concave in price, because the lost value of the option
should be subtracted as a cost of investing. This is the basic message of
the literature on the irreversible investment. Recently, some economists

have attempted to extend this line of research by modeling a firm's

80

81

adjustment costs in a more realistic manner.1 The modeling of taxes in the
context of the irreversible investment framework also provides a promising
research agenda. One interesting extension would be to allow for tax
asymmetries, and scrutinize their interaction with irreversibility.

The absence of a full loss offset scheme is a prime example of tax
asymmetries. Public economist have long studied the relationship between
taxes and risk. Domar and Musgrave (1944) and Stiglitz (1969) are among
the early works. They argue that. if a tax system allows for full loss offset,
the government shares in the risk in a firm’s uncertain future income, so that
risky assets have the same (after-tax) expected rate of return as do the
riskless assets. Gordon (1985) shows in a general equilibrium model that
the risk premium is not affected by the tax, since some of the risk in the
return from an investment is absorbed by the government. The tax system,
therefore, is neutral with respect to risk, in the sense that a firm whose loss
is fully refunded is, in effect, allowed to deduct the cost of risk from taxable
income. To rephrase this argument, firms that operate under an imperfect
loss offset system bear at least a portion of risk, thereby raising the cost of
capital. Auerbach (1986) and Mayer (1986) analyze the effect of no loss

offset on the cost of capital in the presence of carry-forwards and carry-

 

‘ In fact, this research started with a criticism against the q-theory of investment,
in which the functional form of adjustment costs is assumed to be convex and
smooth. The first attack came from Rothschild (1971). who argues that the
concept of convex adjustment costs has a weak foundation in the real world, where
many types of production costs are fixed or actually decreasing. More rigorous
critiques have appeared as a by-product of the theory of irreversible investment.
Most recently, for example, Abel and Eberly (1994) prOpose a generalized model of
investment to encompass both irreversibility and adjustment costs, including (flow)
fixed costs. Caballero and Leahy (1996) forcefully argue that marginal q breaks
down in the presence of (stock) fixed costs.

82

backs of the losses, given the dynamics of investment decisions. They
attempt to show how the effective tax rates are determined, as the firm's
tax positions alter over time. However, their conclusions are not
unambiguous, because the effective tax rates will vary according to the
entire time path of the tax positions of the firm.

In the models discussed so far, the risk is associated with a systematic
risk. or risk-bearing is not a primary interest of analysis. The other line of
the literature perceives the analogy between tax asymmetries and call
options, which is relevant to this chapter. Majd and Myers (1987), for
example, evaluate numerically the effects on a project's net present value of
tax asymmetries with loss carry provisions. In doing so, they use the option
pricing theory combined with Monte Carlo simulation. In their model, the
firm's future tax position is determined endogenously, given the exogenous
future pretax cash flows. Hence the model rules out the possibility that the
future tax positions will alter the current operating decisions which affect
the firm's cash flow.

The present model is distinguished from other option-based models, in
that it allows for irreversibility in investment and operating flexibility in
response to the expected tax changes. The basic structure of the model is
similar to that of MacKie-Mason (1990). However. we employ a different
approach to the firm’s investment decision problem. In the model of
MacKie-Mason, a firm’s cash flow with tax asymmetries is valued by the
contingent claims analysis. a la Black and Scholes (1973). We use an
alternative modeling technique, consistent with the previous chapter;

assuming that the firm is infinitely lived, we set up the value of the firm as a

83

recursive Bellman equation. We also view the government tax claim as a
European call option on the firm's future cash flows.

As will become clear, this approach has two advantages. First, it is
parsimonious, since the solution of the model is solely dependent on the
output price. Second, the options, which affect the firm's investment
decision over the relevant price ranges, are easily identified and valued. The
model will address three aspects of asymmetries: an imperfect loss offset, a
limited tax credit, and irreversibility of investment. We assume a tax system
in which the degree of the loss offset and the investment tax credit differ by
the firm’s profit positions. The firm will then optimize its investment
decision in response to the changes in the net value of the firm that
explicitly take account of the value of the options. The focus of analysis is
to investigate how uncertainty, tax asymmetries, and irreversibility interact
to alter the optimal investment decisions.

In the next section, we present a simple two-period decision model to
capture the flavor of basic features of the full model. In section 3. we
briefly describe the tax system to be considered, and offer an option
interpretation for the tax claim. Section 4 is devoted to the complete model.
After solving the model for the value of a firm and the value of the option to
invest, we compute the optimal triggering prices for investment, and
examine how key parameters affect their values. Some policy discussions

also follow. Section 5 concludes the chapter.

84

3.2. A Two-Period Decision Model

Before proceeding to the full model, we present a two-period decision
model to illustrate some key arguments and results behind it. Consider a
firm that is infinitely lived, and earns the net operating revenue R, at time t.
The firm earns a non-stochastic revenue, R,, in the first period. and a
stochastic revenue, R,, in the second period. Once uncertainty is resolved in
the second period, the revenue remains the same forever. Specifically, the

second period revenue is assumed to be distributed normally according to
R,=R,+a,, where £,~Ni0,a’), so that R,~NiR,,o’/.

The firm pays a corporate tax at rate t that is constant over time. There is
no loss offset for the firm’s operating losses. The firm is assumed to be
risk-neutral, thereby discounting its future cash flows at the riskless interest
rate. r. Finally, we assume that the investment is completely irreversible,
that is, the firm is not able to recoup its initial investment expenditure, K.

Let us first specify the government’s tax claim on the firm's cash flow.
In the absence of a loss offset, the tax claim at any instant can be

expressed as
T, = Maxer,, 0] = rMax[R,, 0], (1 )

which states that the government collects a tax equal to 1R, or 0, whichever
is greater. Therefore, the tax claims are equivalent to a fraction, 1, of the

sum of European call options on the firm’s cash flow. R,, with the exercise

85

price equal to zero.2 T, is an outflow in calculating the firm's after-tax cash
flow. The firm now has a two-period decision problem, which can be solved
by a simple application of dynamic programming. We compute the net
present values (NPVs) from the investing decisions of the first and the
second period, and then compare them to reach the optimal decision; see
Section 1.3 in Chapter 1 for more discussion.

The NPV from investing in the first period is

NP v, = {R,-rMax[R,, 01} + % {EiR,)-rEiMax[R,, 01) } -K

= _:_’ R,-tMax[R,, 01-; rE{Max[l?2. 011M. I2)

We use the fact, EiR,l=R,, to move to the second line.3 If the firm chooses
to wait, it will decide whether to invest, when R, is determined in the
second period.‘ Specifically, if the NPV from investing is positive, it will
proceed with the project. Otherwise, it will give away the investment, and
earn nothing. Therefore, the NPV from waiting, discounted back to the first

period, is

NPV,= l—j;E{Max[-l—t:ii-rlR,-K, 01}. (3)

r

We should observe how asymmetry works between the firm and the

2 If we introduce depreciation allowances, 0,, the tax claim is rMaxiR,-D,, 0], in
which case the exercise price is D,.

3 Notice that Max is a convex function, so the expectations operator should not
pass through the function. Equivalently, EiMax[.]l>Max[Ei.)] by Jensen's
inequality.

‘ We assume here that the revenue depends solely on the output price, which is
stochastic only in the second period. Then the firm observes the price at the
beginning of the second period, determines R,, and decides whether to invest.

86

government for each case. When the firm makes a myopic decision of
investing solely in the first period, the government can take advantage of
contingencies in the second period due to the lack of a loss offset; the tax

authorities receive the tax claim on the firm’s profits, while avoiding the tax

refund on its losses. The term, %rE{Max[R,,0]}}, in equation (2) reflects

this option value. When the firm has the flexibility of waiting, the ball is
with the firm. It will tailor the investment decision to its advantage by
responding optimally to contingencies. After observing the after-tax cash
flow in the second period, it will decide to proceed with the investment, only
if the NPV in (3) is positive. Obviously, the firm's optimal decision is the
one that has the larger positive value between these two choices, NPV, and
NPV,.5

The difference between NPV, and NPV, is the value of waiting, or the
opportunity cost of investing in the first period. This additional cost, which
the traditional approach ignores. must be included in calculating the true
cost of capital for an immediate investment. If the value of waiting is
greater, the firm has a smaller incentive to invest immediately. The
investment in the first period is warranted only when this opportunity cost of
investing is equal to or less than zero. Subtracting (2) from (3), we can

express the value of waiting, W, as a function of R,,

win.) = T17 E{Max[l:—' i1-r)R,-K, 0]} + rMale,, 01

 

5 Notice that, if NPV, <0, investing in the second period is always preferred,
because NPV, is always positive.

87

1+r
r

 

+ {} rEiMale,, 0])-K}- R, (4)

The expression (4) has two expected values of the maximum functions,

E{Max[-l-::i1-r)R,-K, 0]} and E{Max[R,, 0]}. These terms are not easy to

r
evaluate. because they are functions of the stochastic variable, R,. Let us
first consider the latter one. Since R, is normally distributed with mean R,
and variance 0*, and Male,,0] is bounded from below, we ought to use the
censored normal distribution to compute its expected value.“ To do so, we

need to define a new random variable,

3* =0 if R,<0

R*=R2 if R220.

R" is called a lower-censored normal variable. E{Max[R,,0]} is equivalent to

its expected value. ElR‘], which can be calculated using the formula,
EIR *1 =s<DisI +pi1-d>is)) + a’¢is). (5)

where s is a censoring point; p is the mean; ¢(.) is the probability distribution
function (p.d.f.); and CPI.) is the cumulative distribution function (c.d.f.) of
the uncensored variable R,, respectively. Given the censoring point s=0,
the mean p=EiRzl=R,, and the variance 0’, it is straightforward to calculate

EiMale,,0]}. Applying the same procedure, it is not difficult to evaluate

the other term, E{Max[1%r-i 1-r)R,-K, 0]}. Then we can express the value of

waiting. W. as a function of R,, given other parameter values. For a base-

 

“ See Green (1993). pp. 691-3 for a treatment of the censored normal distribution.

88

case computation, we set the parameter values at r=0.05, q=0.2, r=0.3,
3:0. and K = 1.

Figures 3.1 through 3.3 depict W for a range of R,. Figures 3.1 and
3.2 show three separate paths of W in response to changing values of the
tax rate (r) and uncertainty (0') under the no-offset scheme. Figure 3.3
compares the paths of W between the no-loss-offset scheme and the full-
loss-offset scheme for the base case.7 All three figures show that W is
decreasing in R,. We can find the reason by examining the right-hand side
of the expression (4) for W. Notice that all four terms are increasing in R,.
Then W is decreasing in R,, if the difference between the last term and the
first three terms enlarges as R, increases. The last term represents the sum
of the current and the expected pre-tax cash flow from investing in the first
period. The firm would lose this value by deferring the investment. Thus, W
is downward sloping, because the value of the last term increases more
rapidly than those of the first three terms as R, goes up.

The figures also show that W is larger—that is, an immediate
investment is discouraged—when the tax rate is higher, the uncertainty is
greater, or there is no loss offset. This because NPV, in equation (2) and
NPV, in equation (3) are affected differently by the changes in these
parameter values. Let us first examine the tax effect. An increase in the

tax rate reduces both NPV, and NPV,, since it reduces the firm's cash flow.

 

7 We have assumed no loss offset by setting s=0. On the contrary, we can assume
full loss offset by setting s=oo.

89

 

 

 

T=0.3

 

 

 

 

 

 

 

 

 

Figure 3.1 - The Tax Rate It), the First-Period Revenue (R,). and the Value
of Waiting (W)

90

 

 

 

 

 

 

 

 

 

 

 

 

In T T T T T T fr 1 fit I
r- -(
). Cl
)- ..I
VI'I- "I
5 .
II‘ -
\ -.
"' \ U=O.2 ‘
ro- \ ——o=O.3 ‘
k \ __ ~04 .
I-\ \ - U—. a
I. \ _
N- ..
, I
‘—)— -'
)- d
h J
I- ..
O
. “I
T'- L 1 l 4 1 I 1 I l J l L 1 l 4 ‘1
00 01 02 03 04 0.5 06 07 08 09 10

Figure 3.2 - Uncertainty (0'), the First-Period Revenue (R,), and the Value
of Waiting (W)

91

 

 

 

 

I”? T T T T T T f T T m I I “r
)- -I
F u:
N )- .-
\ full loss offset ~
\ — - no loss offset .

 

 

 

 

 

 

 

 

I 1 1 1 1 1 I 1 1 1 1
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
R,

Figure 3.3 - The Loss Offset, the First-Period Revenue (R,). and the Value
of Waiting (W)

92

Recall that the firm that decides to wait has more flexibility than the firm
that decides to invest now in optimizing the investment decision. Then the
higher tax rate would have a smaller influence on the waiting firm than on
the investing firm. This implies that the reduction in NPV, is less than the
reduction in NPV,. and W becomes larger. However, the effect of a tax
increase on W is minimal.

On the other hand, uncertainty has a stronger effect on W than does
the tax rate or the loss-offset scheme, under our plausible parameter values.
In other words, the Jensen effect, or the effect of Jensen’s inequality from
the convexity (concavity) of the value function, is greater than the effect of
the tax rate or the loss-offset scheme. We can explain the importance of
uncertainty in the context of irreversibility. When there is no irreversibility,
we have no value of waiting, so we need only to focus on NPV,.
Investigation of equation (2) shows that NPV, is concave, because the two
subtracted terms, MaxIR,,0] and E{Max[R,,0]}}, are convex. This implies
that uncertainty lowers NPV,. Therefore. if the investment is reversible,
increased uncertainty lowers the firm value, and reduces the incentive to
invest. This is the direct consequence of an option effect.

In the presence of irreversibility, we should also consider NPV, that
contains another option, namely, the option to invest in the second period.
Since equation (3) for NPV, is convex, increased uncertainty raises NPV,. It
follows that, because uncertainty increases NPV, and decreases NPV,. the
value of waiting, W, which equals the difference between these two values,

increases. This implies that, under irreversibility, growing uncertainty

93

further reduces an incentive to invest immediately. In a nutshell, uncertainty
interacts with irreversibility to magnify the value of waiting. Our numerical
solution confirms that this is exactly the case.

There is another salient feature in the uncertainty effect. We can
observe from Figure 3.2 that the effect of increasing uncertainty mitigates at
a higher level of R,. In other words, the vertical distance between the
curves shortens as R, increases. The reason is that the option effects
become negligible, when R, rises to the sufficiently high levels. Recalling

that R,=R,+e,, where e,~Ni0,o’). we can infer that, at a higher level of R,,

R, is less likely to go down enough to be constrained by the options. In
terms of NPV,, the government is less likely to get a chance of avoiding a
tax refund. In terms of NPV,, the firm is less likely to get a chance of
avoiding the down side loss. Likewise, both the government and the firm
lose much of their benefits from flexibility at higher levels of R,. Then the
effect of uncertainty on the value of the options dwindles, making the
options worthless.“

A similar argument can be put forward by examining the effect of the
loss-offset scheme in Figure 3.3. At lower values of R,, W is greater with
no loss offset than with a full loss offset. As R, rises. however, the value of
the options associated with the lack of a loss offset gradually declines. At a
sufficiently high level of R,——in our case. in the vicinity of 0.4—there

remains almost no effect solely from no loss offset, because it would be

 

“ Statistically, this is equivalent to saying that the expected value of a lower
censored random variable decreases. as the censoring region shrinks.

94

very unlikely that R, will go down below zero in the second period. It
follows that the firm incurs an additional cost of investing, if the loss offset
is not available, and the current profitability of the investment is low. This
implies that the absence of loss offset may discriminate against infant firms
or risky firms, because it is highly likely that these firms are in a low profit
position eligible for a loss offset.

The main message of our simple model is that a tax system is not
neutral with respect to risk, and risky investments are discouraged without a
loss offset. There are other studies as well that point out these possibilities.
For example. in the model of Gordon (1985) cited above, there is no risk-
sharing role of the government through income taxes under the no-loss-
offset system. making the tax system biased against risky investments.
Bulow and Summers (1984) articulate another possibility resulting from a
flaw in the depreciation schemes. When the replacement cost of
depreciable capital is fluctuating, the firm bears a capital risk. The firm may
not be able to deduct this cost of risk, since depreciation deductions are
predetermined against the original price of the assets. Then the tax system
discourages an investment in the risky capital.

These models are concerned with systematic risk that cannot be
diversified away by a risk-averse firm. Since we assume a risk-neutral firm,
the firm's risk attitude has no role to play, precluding the risk-sharing effect
of taxes. Instead, our model highlights the option effects that discourage
investment in the absence of a loss offset. The options affect the firm’s
investment decision, regardless of the types of risk. The firm's tax liability

is like selling an option to the government. This option has value, when

95

there is no loss offset. 0n the other hand. the firm has an option to invest
under irreversibility. These two options work together to raise the value of

waiting, and the cost of investing now.

3.3. Tax Asymmetries and Tax Claims: An Option Interpretation

Tax systems in the U.S., as well as in other countries. are characterized
by asymmetries in the tax structure. In many countries, firms' losses can be
carried backward, or carried forward, in order to be written off against
positive profits in other financial years. The refundability of the loss offset
schemes, however, is far from perfect in many cases. It is common that the
number of years over which the loss offset is applied is limited. Firms incur
the interest costs during the carry-forward periods. It follows that firms
investing in risky projects, or in their early years of operation, are likely to
pay higher taxes. Asymmetries can also occur on the credit side. Firms are
allowed to offset their taxes payable with the investment tax credit. But
there are usually some caps on the amount of credit that is available.“ Firms
are likewise treated differently, depending on whether they are subject to a
credit limit. For the income levels under which the credit ceiling is binding,

firms are, in fact, taxed at a lower marginal tax rate. since the tax credit is

 

° In the U.S., the investment tax credit was repealed by the 1986 tax reform. Until
that time, the tax credit had been awarded with a ceiling. In 1981, a form of
refundability was enacted. through 'Safe Harbor Leasing", to provide tax benefits
for firms that could not fully use accelerated depreciation because of lack of
sufficient tax liability. But this was repealed in 1982.

96

awarded as a fraction of taxes payable.

Consider now a hypothetical asymmetric tax system. A firm pays
income tax at rate r on its stochastic net operating income, R,. The firm is
allowed a deprecation deduction, 0,, so long as the operating profit is
sufficiently large to be deducted. We assume that D, is predetermined,
while R, is contingent on the states of nature. The firm is also eligible for
the investment tax credit at rate x, but the amount cannot exceed 1009

percent of the taxes payable.1o

When the firm's taxable income, R,-D,. is
negative, the tax authorities compensate the firm for its loss. The amount
of tax refund is equal to a certain fraction, 17, of negative taxes payable.
Hence. the degree of the loss offset depends on the value of n.

Algebraically, the tax function is expressed as a composite maximum

and minimum function of R,,
TiR,) = Max[{riR,-D,)-Min[9riR,-D,), xKI}, at! 1-9) iR,-D,)]. (6)

As the nested functional form implies, this function has two kinks. As
illustrated in Figure 3.4, the kinks occur at R, (=D.) and R,(=xK/r6+D,),
respectively. When R, is less than R, (region 1), the firm receives a refund
equal to nrii-HliD,-R,). When R, is between R, and R,, (region 2), it pays

taxes equal to ti 1-9liR,-D,l.

 

‘° We assume that the investment tax credit is allowed evenly throughout the
lifetime of the firm. We can think of it as a credit being given against the interest
cost on the investment expenditure, rK, in each time. Then its capitalization value

Q
is approximately equal to the initial cost of investment, K, because rKZ (l )t
+r
(=0
where r is the interest rate. This assumption is necessary in order to make the
investment credit occur continuously for a firm that invests in a single stand-alone
project.

 

3K,

 

97

 

Tax
put in
region 2
put in
region 1
\ 0
R/
nr(l-9)(R-D) ,’
I
3'. ’
O. ’
I
I
I

 

can in T(R'D)'KK

region 2
.,.r(1-0)(R-D)

 

 

Revenue (R,)

call in
region 3

 

 

region 1: R, < R,
region 2: R, s R, <R,
region 3: R, 2 R,,

 

 

Figure 3.4 - Firm’s Revenue and Tax Claim

- *1.—

98

Finally, R, is sufficiently high to exceed R,, (region 3), the amount of taxes
payable is riR,-D,)-xK. Note that the credit ceiling works in region 2, but it is
no longer binding in region 3, because the firms has a sufficient taxable
income for the full credit to be awarded. The tax function has three
different marginal tax rates across tax brackets.

When there are tax asymmetries and the firm’s future income is
uncertain, the tax claims are equivalent to perpetual European options on a
stream of the firm’s expected future cash flows. The options are a
composite, in that they involve both a call option and a put option. The
shaded areas in Figure 3.4 indicate regions that correspond to each option,
to be explained below. When in region 1, the government has an option
from the perspective of no offset to receive ril-PliR,-D,). if R, goes up into
region 2, or refund nothing, otherwise. It is exactly like having a fraction of
a call option on an asset to be exercised at time t, where the asset price is
R, and the exercise price is D, at time t.

When in region 2, the government has two options. One is a call option
that results from the credit limit applied to region 2; it has an option to
receive an additional amount, 9iR,-D,)-xK, if R, goes up further into region 3,
or receive no additional amount, otherwise.11 The other is a put option that
stems from the imperfect loss offset; it has an option to receive iI-nlrii-
alliD,-R,), if R, goes down into region 1, or receive nothing, otherwise. Let

us examine the nature of the put option further. From the perspective of full

 

" MacKie-Mason (1990) refers to this additional tax claim as 'cream-skimming’. If
xK= 9iR,—D,l, or when there is no credit limit, the call option—or the cream-
skimming region in Mackie-Mason's terminology—disappears.

99

loss offsets in region 2, the government has to refund a full amount, ri1-
I9)(D,-R,), for the firm’s operating loss. When loss offset is imperfect,
however. the government need not refund fully. It saves a fraction, (74]).
by refunding only a fraction, )7. Thus, the government has an option to
receive i1-r7lri1-0)iD,-R,) that is not refunded. It is analogous to having a
fraction of the put option on an asset to be exercised at time t, where the
exercise price is D, and the asset price is R,. 1-17 is the portion of the put
option the government can receive when exercised. As mentioned above, it
also indicates the degree of a loss offset to be given to the firm. When 7)
equals zero, the put option is exercised for the firm’s entire revenue, so
there is absolutely no offset.12 0n the other hand, when 1) equals one, there
is no put option, so the government gives a full loss offset. By the same
logic. we can identify a put option in region 3 for the government to receive
xK-OiR,-D,l that comes from the credit ceiling. Note that the put option in
region 3 is a mirror image of the call option in region 2.

It is obvious from our discussion that tax asymmetries are directly
associated with options. Therefore, the correct evaluation of tax claims

should be based on the firm recognition that the government has the options

 

‘2 Ignoring region 3, this is exactly the case the earlier two-period model describes.
In that model, we interpret no loss offset as the government having a call option.
From the perspective of a full loss offset, this is equivalent to having a put option;
if R, falls below R,, the government can earn 0, in exchange for R, by exercising the
put option, when the tax factor is suppressed. Algebraically, this relationship is
expressed as MaxiR,—D,. 0]=R,-D,+Max[D,-R,, 0]. The maximum function on the
left-hand side represents a call option, while the maximum function on the right-
hand side is a put option. The remaining term, R,-D,, represents the relationship for
a full loss offset. This is called the 'put-call parity' in the finance literature, which
says that. at the time of expiration, the value of a call is equal to the value of a
portfolio consisting of the underlying asset, borrowing money equal to the exercise
price, and a put option.

100

on a stream of uncertain future cash flows of firms. To capture the true
value of the firms’ tax liabilities, it is important to identify correctly the
options involved, and trace carefully how their values alter, as uncertainty
evolves over time. We should also remember that there may be other
options, caused by other asymmetries firms are facing. These hidden option
values may constitute a substantial portion of the whole firm value, and

must be taken into consideration for the optimal investment decisions.

3.4. The Model of Dynamic Investment Decisions

3.4.1. Reversibility and the Value of the Firm

In the two-period decision model presented earlier, the firm's problem is
about the optimal timing of investment between two points of time; that is,
whether to invest now or to wait and decide in the next period. We now
develop our model further by extending the firm's horizon on decision-
making, and enriching the tax structure. The model will serve two purposes.
The first is to recast the firm's optimal investment problem into a
continuous-time dynamic programming framework, and solve it for the value
of the firm that explicitly includes the value of the options. The second is to
determine the threshold level of the output price at which launching an
investment is optimal.

We closely follow the modeling strategy of Chapter 2 in production
technology and the price process. A representative firm is considering a

single discrete investment project. The project can be undertaken at any

101

time at an initial sunk cost of K. Once in operation, it produces one unit of
output indefinitely with no operating cost. As an additional element of the
model, we introduce capital depreciation that follows a Poisson process; at
any time t, the firm faces a probability, Mt, that the project will collapse
and cease production during an infinitesimal period of time, dt. The Poisson
process, denoted by q, has a value of one, if the project dies; otherwise, it
has a value of zero.13

The output price is assumed to be given exogenously, and follow the

geometric Brownian motion,

dP= ant-I- anw, (7)

where dw is an increment to the standard Wiener process. With these
assumptions, the firm’s operating profit follows a combined geometric

Brownian motion-Poisson process.

dP = ant + a'Pdw-qu, ( 8)

where dq is an increment of the Poisson process with mean arrival rate 2..

dq equals either one with the probability of Adt, or zero with the probability
of (1-Mt). We can interpret equation (8) in the following way: over the
infinitesimal interval of time dt. there is a small probability Mt that the profit
flow will stop forever, or it will continue to evolve according to the

geometric Brownian motion, ant+anw. We assume that the firm is risk-

 

‘3 Thus, we are assuming exponential depreciation most commonly used in
investment theory. In the deterministic context, this is equivalent to assuming
geometric decay in asset efficiency, because the expected value of the firm reduces
geometrically. See Dixit and Pindyck (1994), Chap. 6, for a discussion of the
Poisson process as a form of depreciation.

102

neutral and attempts to maximize its expected future cash flows, discounted
at the after-tax interest rate. Finally, the firms pays the corporate tax on its
operating income. We assume that the structure of taxation is the same as
the one modeled in equation (6) above.

Combining the firm’s production technology and the tax scheme, we

characterize the firm’s optimization problem as

ViP,) = mpax E, I: (P, - TiP,))e’"'"‘“ds, (9)

where TlP,) =Max[{tiP,-D,)-Min[92'iP,-D,), xKJ}, nrii-GIIP,-D,)].
TIP.) has the same functional form as (6). except that P, is substituted

for R,. The recursive Bellman equation for this problem is
ViPl =[P- TiPlldt+ e"""""E[ViP-I- dP)]. (10)

In the spirit of the dynamic programming principle, the first term on the
right-hand side is the immediate cash flow from time 0 to dt, and the second
(term is the expected value of the firm from dt to infinity. Using the fact

that e"”"""z 1-rI1-rldt and EidVl =E[ViP+dPl]-V(P). we obtain
i7-r)rViPIdt=[P-TiPl]dt-I-EidV). (1 1 )

This equation is an intertemporal arbitrage condition; see equation (18) in
Chapter 2 for its interpretation.

We need to apply Ito’s lemma to expand dV, since ViPl is a function of
a random variable. We have two terms in EidV), each of which represents
two different stochastic processes. For the geometric Brownian motion

process. we have the familiar expression.

103

EidVBI = aPV’iPldt-l- éo’P“ V'iPldt. (12)

To derive the corresponding expression for the Poisson process, recall that
dq equals either one with the probability of Mt, or zero with the probability
of (1-Mt). This implies that P jumps to zero with probability Mt when the
project dies, and P continues to evolve when the project survives. Thus the

expected change in dV attributed to the Poisson process is

E! d Vp] = ldtIViOI- VIP/I + i 7-MtIIVIPl- VIP/J = vi. ViPldt. (1 3)

Combining (12) and (13), we have

EidV) = aPV’iPldt + -:-O'2P2 V'iPldt-fi. ViPldt. (14)

Substituting (14) into equation (11), eliminating dt and rearranging, we get a

second-order differential equation,
% 02P2 V'iPl + aPV’iP/-[i1-r)r+ 21 WP) + [P- TIP)! = o. (1 5)
We can rewrite this equation as a risk-adjusted form,
ZI' 02P2V'iPl + ii-rlir-6IPV’iPl-[i1-rlr-I- .uvrP) + [P- TIP)! = o, (1 6)

where 6 represents an opportunity cost of delaying the investment.“
Solving equation (16) for the complementary function is

straightforward, which gives

 

“ See the related discussion in Section 2.3 of Chapter 2. We assume here that the
project risk associated with the Poisson jump is fully diversifiable, so that no
adjustment is necessary for the Poisson risk.

104
V"iP}=/I,P“I +A,P“’, (17)

where

 

fl,=-27—-i1-tlir-6)/oz+ Jm - rm -5) / 0’2 - 1,12 +2rr— rm +2) /0'2 >1, (1 a)

 

fl.» =é-i1-tlir-6l/02-Jii1- rlir - a) / 0’2 — 1,12 +2n- up + 2.) /0'2 < o. (19)

and A, and A, remain to be determined.

Solving for the particular solution is somewhat cumbersome due to the
presence of the complicated tax function, TiPl. From our discussion in the
last section. we can easily infer that, when time subscripts are suppressed.

TiP) is written as

rr7111-6UiP—D) if P<P,.

77p}... I 1(1— 9MP —0) if P, S P < P,, (20)

 

kriP-Dl—KK if P 2P,.

To derive the particular solution, WiP). we plug a trial solution
V'iP)=B,P+B,D+B,K into equation (16), and compare term by term to get

the parametric expressions for 8,, B, and 8,. By working each case through.

 

 

 

 

weobtain
r .
Wzli-nrli—911p+flffl:_gtp ,-, p<PV
(co-III) 60
Vp,p,=,vn=”'“7’“”P+“"‘“’o if PsP<P (21)
2 (CU-(ll) a) 7 2'
\éP___ (1_T)p+_€_D+£.K if P2P2,
\ {CD-W a) a)

 

 

105
where w=ri7-r)+). and w=(1-r)ir-6).
Finally, combining equation (17) for VIP) with equation (21) for VIP) yields
the complete solution for the value of the firm,
'v,(P) = A,P" +A,P“= + v," if P < P,,

v/p) =<V21PI= B,P“' +B,P”= +v; if P, s P <P, (22,

 

V,iP) = C,P“ + C,P“* + v; if P 2 P,,

The interpretation of complementary functions of this equation is
straightforward from our discussion on tax claims as the options on the
firm’s uncertain future cash flows. The first two terms of V,iP) represent a
call option to the government. This option may be exercised for an
additional tax, if P goes up above P,. To rephrase it. the option will not be
exercised when P is very low, and eventually the option has no value as P
goes to zero. Recalling that ,6,<0. this implies that A, should be zero, since
the second term, A213”: , goes to infinity as P goes to zero.

In the second equation, V,iP), there are two options; when P rises
above P,, a call option may be exercised to get an additional tax. When P
falls below P,, a put option may be exercised to prevent a full refund. Using
again the limiting argument above, we can infer that the value of the call
option goes to zero as P goes to zero, while the value of the put option goes
to zero as P goes to co. Recalling that ,B,>1 and ,B,<0, this implies that the

first term, B,P’SI , is for the call option, and the second term, B,P" , is for the

put option.

106

The third equation, V,iP), involves a put option. The option has value,
because the ceiling on the tax credit is binding and benefits the government.
if P falls below P,. The same limiting argument stipulates that the term,
C,P‘“l , should disappear, only leaving CZP‘z.

We now have four constants, A,, 3,, 8,, and. C,, which are to be
determined by imposing appropriate boundary conditions. The necessary
conditions are the value-matching and smooth-pasting conditions at the two
boundaries of the tax function, P, and P,.‘“: the primitive functions and their
derivatives with respect to P, both of which are evaluated at the boundaries,
should have the same values for a set of neighboring functions. In our
parametric specification of the tax function, P, and P, are equal to D and

xK/10+ 0, respectively. Imposing boundary conditions on equation (22) gives
A,Dfi' + (1‘, +H,)D = 8,0’3’ + 8,052 + (1‘2 +£1le (value-matching for V, and V2)
fi,A,D’“"” +I‘, = fi,B,D’“"" +5,B,D’“2"’ +1‘, (smooth-pasting for V, and V,)
B,A“' +B,A”2 +1‘,A +H,D = 0,7151 +1'3A +H30+13K (value matching for V, and V3)

)6,B,A““"" + p,a,AE'fiz-“ + r, = ,6,C,A”“2‘” + r, (smooth-pasting for v, and v3).

= [1- nth—(9)]

where A = KK/2'9+ D, F,
(a) - I/Il

 

1.2:{1-ri1-611’ 1.3: i1-rl .
(co-1)!) (to-Ia)

 

= ”—1), H2: 111—9),
(0.7-VII a)

 

 

I

 

‘“ We have briefly discussed these conditions in the last chapter. See Dixit (1993)
for an introductory account of restrictions imposed for the optimal control of
stochastic process.

107

, and 1,: 5. (23)
(0

Solving this equation system for A,, 8,, 8,, and C,, we finally obtain the
value of the firm, VIP). We should notice that all four constants must have
negative signs. The reason is that it is the government that has the options
of value, so they are negative factors in the value of the firm.

The value of the firm, VIP), is affected by a set of combinations of
parameters. We examine how VIP) is sensitive to a change in the tax
system, by hypothesizing three tax regimes. The first regime is
characterized by a neutral system with no tax credit and a full loss offset
((9: 0, x=0 and r7: 1). It is well established in the neoclassical theory of
investment that a tax system is neutral with respect to the investment
decisions of a firm, if the capital expenditure is expensed. To derive the
neutrality condition formally in the first regime, we need to determine the
value of the firm under the neoclassical assumption of complete reversibility.
For a simple exposition, assume that a firm is risk-neutral and there is no
growth trend in P,, so that r-6=a=0. The expected present value of net

cash flows during the time interval between 0 and t is

t _ _
E I [P - II P — D))] e”"”’“ds = P “(P D) (l —e""""t ), ( 24)
0 r(l — t)

 

This is the value of the firm, if the firm’s project continues to yield the cash
flow until time t.
Since we model depreciation by a Poisson process with mean arrival

rate A. it is not too difficult to show that the probability of the firm's cash

108

flow fluctuating until it suddenly drops to zero at time t is 29“; see Freund
(1992), Chap. 6, for a derivation. The value of the firm during its entire life,

then, is given by

P-r(P-D) _,,,_,,, _PI1-r)+rD
r(1—r) (I e )0“- ri1—r)+.i '

 

 

VIP): fie-‘1 (25)

We can obtain the same result by plugging the associated parameter values,
6: 0, x=0, 77:1, and r-6= 0. into equation (21), since we need not consider
options (complementary functions) under the neutral tax system. In equation
(25). 10 is a value term from depreciation deductions. If we set D=ilK/t.
this term equals .1K. This is equivalent to expensing of the capital
expenditure, K, through depreciation deductions, since .1 can be regarded as
the mean depreciation rate. Let us examine the traditional optimal

investment rule that requires VIPl=K, or from (25).

Pi1-t}+1’D-
rI1—t)+1

 

=0. (26)

If we assume that the capital expenditure is expensed. we can set D=)1K/t

to get

i1—rliP—rKl
ri1—r)+l -0 (27)

 

If P=rK, or the price is equal to the cost of capital, this condition always
holds regardless of r. The tax system is neutral with respect to investment
in the neoclassical sense.

The remaining two regimes deviate from tax neutrality, due to both the

lack of the loss offset and the presence of the investment tax credit. The

109

second regime symbolizes a partial-tax-credit-cum-partial-loss-offset system
(6:0.5, x=0.008 and q=0.5), whereas the third regime represents a full-
tax-credit-cum-no-Ioss-offset system (9:1, x=0.016 and n=0).‘“

Figures 3.5 and 3.6 describe how the value of the firm, V, evolves as a
function of P under these three tax systems. Each figure is drawn for
q=0.2 and 0.4, respectively. We set other parameters at r=6=0.04,
r=0.3, A=0.01, K=100, and D=71K/r. Notice that the capital expenditure,
K, is expensed through the depreciation deduction, D. In Figure 3.5, the
solid line shows the path of V for the neutral tax regime. It starts with a
positive value. and rises straight over the range of P. The firm has some
value even at P=0. since it receives a refund through depreciation
deductions for its operating loss. The value line is straight. because no
option is involved under this system. The dashed and the dotted lines are
for the partially neutral and the non-neutral systems, respectively. They

have lower values at P=0. since the loss offset is imperfect. The shapes of
the graphs are slightly concave. because the options affect the firm value
negatively. As P rises to a sufficiently higher level, the non-neutral paths
outpace the neutral path. The reason is that, as P goes up. the positive
effect of the tax credit increases, while the negative effect of the imperfect
(no) loss offset decreases. Once P reaches a certain high level, the value
addition from the credit availability dominates the value reduction from the

incomplete loss offset.

 

‘“ As discussed in footnote 10, the investment tax credit is assumed to be given
against the interest cost on investment, rK. Hence. I: is calculated by multiplying
the pure tax credit parameter by the interest rate, r.

110

 

 

no credit—full offset //

 

— - portiol credit-partial offset
--- full credit-no offset //
/

 

 

 

V(P)
150 200 250 300 550 400 450

100

50

 

 

 

 

18 20

Figure 3.5 - The Tax System, the Output Price. and the Value of the Firm
(a=0.2)

111

 

 

 

 

 

 

 

 

 

o
:2.) T T T T r r TT 1 T1 T T T T T
8
V. I?
/

o no credit—full offset
ID - . . .
In -— - portiol credit—partial offset / “I
o —-- full credit—no offset ’
O -I
pf)
S

A 7
N

D.

v O

> o -
(\l
o
0 cut
0
O -I
O .
In
D L 4

18 20

Figure 3.6 - The Tax System, the Output Price,
(a=0.4)

 

and the Value of the Firm

112

At a very high level of P, both effects lose most of their power, and the
three paths run parallel to each other.

In Figure 3.6, we increase the level of uncertainty and reproduce the
paths. The starting points are almost the same as in the previous picture.
Yet adding uncertainty has the effect of shrinking the gap between curves at
the higher values of P. Why is this so? A careful observation of the graphs
shows that the neutral path does not alter, whereas the non-neutral paths
shift slightly downward. This is the consequence of the option effects. As
we have discussed, the options are worthless under the neutral system.
This is not the case for the non-neutral system. As uncertainty increases,
the value of the options also increases. This implies that, under the non-
neutral system, the value of the firm is more adversely affected at a higher
level of uncertainty and at a higher level of price. The two figures also
show that the non-neutral tax system discriminates against unprofitable
firms, since they are more likely to be injured by the imperfect loss offset.
This is the same point that we have made in the two-period model. The
difference is that the full model has a compound option, while the two-
period version has a simple option.

Another noticeable feature of the figures is that the curves are close to
each other at both levels of uncertainty, and that they do not shift very
much as uncertainty increases. This is as much as to say that neither the
tax system nor uncertainty has a strong effect on investment in the
neoclassical framework with reversible investment. To make the argument

more convincing, we calculate the critical level of P, at which VIP)=K, or it

 

113

is Optimal to trigger an investment. This is the optimal condition of the
naive net-present-value method.

In Table 3.1, we show how the optimal price responds to the changes
in the tax rate (r) and uncertainty (0) under the three tax systems. In the
neutral system, the optimal price is P*=rK=4.0 at any level of r and 0'.
Neither the tax rate nor the degree of uncertainty have an effect on
investment decisions.

In the partially neutral and the non-neutral systems, these two
parameters affect the optimal price in opposite directions; an increase in
uncertainty raises P", while an increase in the tax rate lowers P’. As we
have seen in Figures 3.5 and 3.6, increased uncertainty raises the option
value, V‘, which, in turn, lowers the value of the firm, V. Hence P‘ should
rise to compensate that value reduction. On the contrary, an increase in the
tax rate raises V and lowers P“ through two effects. First. the resulting
increase in the immediate tax liability reduces the fundamental value of the
firm, V”, so the value of the firm decreases. Second, the tax rate increase
reduces the value of [3, and the (absolute) value of the option, Vc, decreases
accordingly. Since Vc is a subtracted component of firm value, the value of
the firm increases.

In our parameter setting, the option effect is greater than the tax rate
effect, thereby raising V and lowering P*. The overall picture of their
effects, however, is not impressive. Taking P*=4.0 as a benchmark, all the
threshold prices fall within the range of 3.5 to 4.5. As we will see below,

this is not the case, when there is irreversibility in investment.

114

Table 3.1- Uncertainty, Tax Rate and Optimal Price for Investment
(Reversibility Case)

 

0 Neutral Partial Non-neutral

 

0.1 4.0 3.9 3.8

 

 

 

Source: Author’s calculations

3.4.2. Irreversibility and the Option to Invest

In our discussion thus far, we have argued that tax claims are in fact
the options on a stream of the firm’s uncertain cash flows, and that the
value of these options must be considered in assessing the firm value. We
have also derived the optimal investment rule by valuing the firm’s net cash
flows in an asymmetric tax system under the reversibility of investment. If
the irreversibility of investment is imposed on our model. another important
consideration is in order; the optimal investment rule requires that the option
to invest be a part of the model solution. We have shown in the two-period
model that. if an investment is irreversible, the option to invest in the next
available time has value, so an immediate investment should include this lost

value of the option as an additional cost.

115

It is useful to distinguish the firm’s option to invest from the
government's option to tax by their option concepts. The tax option is
exercised only at the time the firm's operating revenue is determined,
whereas the investment option can be exercised at any time the firm
believes that it is optimal to do so. In other words, the firm has a complete
discretion in undertaking an investment. In the option terminology, the tax
claim is an European option, and the option to invest is an American option;
see the discussion in Section 1.1 of Chapter 1.

The value of the option to invest stems from the expected gains from
the price changes for an investment that has yet to be implemented. Hence,
it evolves according to an equation of motion similar to equation (16), with a

difference being that it has no immediate cash-flow term,
FIP) = e"”"""E[vrP + dP)]. (28)

Manipulating this equation through the same steps as before, we obtain a

differential equation,

% a’P’F'lP) + aPF’IPI-[l 1 -r)r + AJFIP) = 0. (29’

Since this is a homogenous form. the solution for FIP) has a simple form,
F/P;.—.D,P" +02P‘2. (30)

This option to invest has zero value as P goes to zero. By applying the
limiting argument discussed earlier, we can eliminate the second term, and

are left with

HP) =D,P”'. (31)

116

The new investment rule requires that, by investing immediately, the

firm gains VlPl-K in exchange for FIP), instead of for nothing. To determine

the threshold price, P", of investing now, we again have recourse to the

arbitrage principle by imposing the value-matching and smooth-pasting

conditions at P*. For each price range of equation (22), applying the

conditions to each pair of VlPl-K and FIP) gives a set of the following

equafions:

filHP<P,
AIP'I“ +1“,P' +H,D-K = 0,1107“
filAilP'I’A‘“ + 1‘, = golf/W“
(ii) if P,SP<P2
311V!“ +leP'/”’ +F2P' +H20-K = 0100')“
flIBilplllfi-fl +,BZBZ(p';U32-H ”—2 = flDflP'I’fi’”
(iii) if P.>_P2
C,(P'l’3’ +1‘3P' +H30 + 13K— K = Dflp'fl:

mare/”’2‘” + 1‘, = 13,0,IP'1’4‘“

(from V,IP*I-K=F(P*ll

(from WP”) =F’IP*Il.

(from V2(P*)-K=FIP*I)

(from VZ'IP'I =F’(P*ll.

(from V3(P*I-K=F/P*l)

(from vgrP'I =F’IP*ll.

(32)

where 1“,, 1‘2, 1'3, 11,, H2, H3, and 13 are as defined in (23). Note that A,, 8,,

32, and C, are already determined from solving equations in (18). We solve

 

117

each set of equations for P* and 0,, and pin down the P" that is in the
corresponding price range.

As we did in Table 3.1, we show in Table 3.2 how the threshold price,
P*, responds to the changes in the tax rate (r) and uncertainty (0') under the
three tax systems. When we compare the two tables, we can notice a few
distinctions. The first one is that the levels of P‘ are much higher and have
a large variation in Table 3.2. Using P* =rK=4.0 as a benchmark, which is
the threshold price in the neoclassical neutral tax system, we can easily see

that P* varies widely from 5.8 to 16.2 in Table 3.2.

Table 3. 2- Uncertainty, Tax Rate and Optimal Price for Investment
(Irreversibility Case)

 

 

 

 

 

0' Neutral Partial Non-neutral
0.1 5.8 6.1 6.1 (6.0)
0.3 11.8 11.7 11.2(13.2)
0.4 16.2 16.2 15.3 (18.6)

T

0.2 8.0 8.6 8. 8 (8. 7) H
0.4 8.9 8.2 7. 5 (9. 5)
0.5 9.5 8.1 7.1 (10.4)

 

Source: Author's calculations
Note: 1. We set o=0.2 and r=0.3 as a base case. The values of other
parameters are as explained in the text.
2. Figures in parentheses are for a tax regime with no loss offset and
no tax credit.

118

The variation is between 3.5 to 4.5 in Table 3.2. This is a good evidence
for the argument that the option to invest has a substantial effect on the
investment decision. Needless to say, both the higher values of and the
larger variation in P* are attributed to the presence of a complete
irreversibility. Irreversibility of investment, combined with uncertainty, gives
rise to a large gap between the neoclassical result in Table 3.1 and the new
result in Table 3.2.17

The second distinction is that uncertainty contributes more significantly
to investment decisions than does the tax rate. In Table 3.1, the difference
in their effects between uncertainty and the tax rate is not discernible. We
observe in Table 3.2 that the uncertainty effect easily overrides the tax
effect across all tax systems. For example, from the base case of r=0.3
and a'=0.2, adding uncertainty by 0.1 raises P‘ by 3.4, whereas adding the
tax rate by 0.1 raises P* by mere 0.5 in the neutral system. As discussed in
the reversibility case, a tax increase lowers P" in the non-neutral system,
because the option effect dominates the tax rate effect. This is also the
reason why shifting from the neutral to the non-neutral system lowers P‘.
Note that, if we ignore the tax credit and consider the no-Ioss-offset case
only, a rise in r raises P*, even though its effect is minimal in our tax

specification (see the figures in parentheses). Finally, we have argued that,

 

‘7 In the last chapter, we deal with a model with varying degree of irreversibility,
and produce less dramatic—yet still significant—results with a small level of
irreversibility. Once we accept the fact that irreversibility pervades real-world
investment, it stands to reason to cast doubts on the robustness of the complete
reversibility assumption of neoclassical theory of investment. Other factors, such
as the firm's risk preference, can combine to magnify the effects; see Chapter 4 for
a model addressing the attitudes of a firm toward risk.

119

in the neoclassical world, both the tax rate and uncertainty are neutral in
their effects on investment decisions, provided that there is full loss offset
and no tax credit, and the capital expenditure is expensed. This is not the
case, once irreversibility is taken into account. P‘ alters significantly in
response to a change in the tax rate, as well as to a change in the degree of
uncertainty. We should, therefore, be cautious in evaluating tax neutrality,
when an investment involves Substantial irreversibility and uncertainty.

'We recapitulate what our discussion implies for tax policy: (i) The
optimal threshold price to initiate an investment differs both in terms of
levels and variations between the world of reversible investment and the
world of irreversible investment. The option to invest plays a fundamental
role in raising the threshold price, and the tax neutrality in the neoclassical
sense no longer holds under irreversibility:

(ii) In the neoclassical model, there is almost no difference in their
effects between uncertainty and the tax rate. When irreversibility is
introduced, uncertainty has a tremendous effect on a firm's investment
decisions in the model. The tax effect is dominated by the uncertainty
effect;

(iii) The tax rate increase works in opposite directions between the
neutral and the non-neutral system under irreversibility. This implies that tax
policy may have a limited effect on investment, or it may be difficult to
predict its effect, when the tax system is characterized by a hybrid of tax
incentives (e.g., the tax credit) and tax disincentives (e.g., the lack of loss
offsets). On the contrary, an increase in uncertainty works in one direction

across tax systems, which implies that, if irreversibility problem is serious, a

 

1'

 

120

reduction in uncertainty in price and policy may have a stronger impact on

investment.

3.5. Summary and Conclusions

A rational firm would base its investment decision on both the current
and future profitability of a project under consideration. When the firm is
restricted by asymmetries and future profitability is uncertain, the firm’s
forward-looking decision must take account of the options involved in
undertaking it. We have modeled asymmetries through an asymmetric tax
system and complete irreversibility of investment. Tax asymmetries give the
government an option on the firm's future profits, which has an adverse
effect on the value of the firm. Irreversibility offers the firm an option to
invest at the best available time, which constitutes an opportunity cost of
investing now. These two options, therefore, raise the cost of capital for
the firm that undertakes an immediate investment.

An incomplete loss offset is a typical tax asymmetry that prevails in
most countries. The two-period decision model clearly shows that the
absence of loss offsets discourages investment, especially for less profitable
firms. This is the consequence of the option effect argued in other similar
studies (e.g., Majd and Myers (1986), and MacKie-Mason (1990)). In this
tradition of work, the option effect has nothing to do with the firm’s risk
preferences, since the firm is assumed to be risk-neutral. Thus, uncertainty
plays a fundamental role, regardless of whether it is associated with market

risk or specific risk. There is another line of literature, which exclusively

 

121

deals with market risk (e.g., Gordon (1985)). These studies show that the
absence of risk-sharing through taxation discourages investment, when the
firm behaves as if it were averse to risk. The option-based model can
address both issues in an integrated framework.

According to numerical solutions under plausible parameter values, the
option to invest that stems from irreversibility has significant value. Driven
primarily by this option effect, the cost of capital deviates to a great extent
from the levels the neoclassical analysis suggests. Increased uncertainty
significantly amplifies the negative influence of the options on investment.
The tax effect is overridden by the uncertainty effect, both in magnitude and
predictability across tax systems. The traditional tax neutrality result is no
longer valid. When irreversibility of investment is seriously involved,
economic stabilization programs, including transparent and consistent policy
stances, may be more effective than tax cuts or other tax incentives in
stimulating investment.18 The welfare consequences of tax policies need to

be evaluated with due regard to the effect of irreversibility.

 

1" As an evidence of this argument, the last fifteen years have witnessed reduced
variability in the inflation rate, when compared with the previous fifteen years.
This might have a beneficial effect on investment.

Chapter 4

Asymmetric Information, Risky Investments, and
Financial Market Fragility

4.1. Introduction

None of the economic fields has been more affected by the theory of
imperfect information than the analysis of financial markets. Many studies
have contributed to important theoretical results that address market failures
and problems stemming from asymmetric information between lenders and
borrowers. A typical informational asymmetry found in financial markets is
that borrowers have better information about the profitability and riskiness
of a project under consideration, than do lenders. Asymmetries in
information create certain features of financial markets, sometimes
combined with other asymmetries. A good example is credit rationing, the
classic model of which was pioneered by Stiglitz and Weiss (1981).

The modeling strategy based on asymmetric information also proves
useful in explaining numerous banking problems that have occurred in both
developed and developing economies over the last two decades. One recent
example is the Mexican financial crisis in 1994, and, at this moment,
selected Asian countries have been undergoing a series of banking and
currency crises. The financial crisis in Asia began in Thailand in the summer

of 1997, and swept through Indonesia, Malaysia, and Korea throughout the

122

123

rest of the year.‘ The Asian episode seems to share common characteristics
with those that have happened in the past. Yet a distinctive feature should
be noted. Prior to the onset of the crisis, the economic performances and
policy stances of the Asian countries were not bad enough to lead to
warnings against any immediate crisis. These countries had sustained good
fiscal balance, moderate inflation, and low unemployment, albeit a fair
amount of the trade and current account deficits. Nevertheless, once the
crisis emerged on the surface, it has had a tremendous effect on the entire
region, as well as on the individual economies. This observation has led
economists to reexamine the Asian crisis, both from old and new
perspectives.

Economists have long sought to investigate panics and crises in the
financial markets. Until recently, banking problems and currency collapses
had been studied as a separate agenda. Yet fundamental changes in the
international financial system, including globalization of financial markets
and advances in financial innovation, have made these two phenomena
indistinguishable. We can divide current research efforts to study financial
crises into three categories. The first approach is to examine the

microeconomic problems that are deeply rooted in the economy, and that

 

‘ In some respects, the Asian economic crisis has prompted an interesting research
agenda. The spread of the crisis was so pervasive and severe that nobody
anticipated the economic depression of this sort. Even though there have been

_ some skepticism and warnings about the so-called Asian miracle, the skeptics have
primarily pointed toward a gradual erosion of long-term growth. For example, in his
provocative article, entitled "The Myth of Asia’s Miracle,” published in Foreign
Affairs in 1994, Krugman expresses a pessimistic view on the future growth
prospects for the Asian economies. He cites as supporting evidence several papers,
including Kim and Lau (1994), and Young (1994), who argue that productivity
growth is not the major source of Asian growth.

124

provide the seeds of a banking crisis. These studies emphasize such
behavioral and institutional factors as informational asymmetries, and their
macroeconomic consequences that lead to some initial conditions for the
crises.2 The second literature focuses on modeling of the banking and
currency crisis. There are several original models and numerous variants of
them in this area. The first generation of these models investigates the
currency crisis under the fixed-exchange-rate system. Later studies shift to
the analysis of the financial crisis. These studies encompass the entire
financial sector, especially in the emerging markets.3 The final, and the
most recent, approach is to investigate the transmission of a crisis among
sectors and between economies. These models use intertemporal general
equilibrium analysis to show how shocks are propagated throughout the
economic system, and to reproduce stylized facts of the crisis. A bit scanty
as it may be, this literature provides a useful insight into a contagious effect
of the crisis—what came to be called the ”tequila effect" named after the
Latin American Crisis in 1994-5—which is still unresolved among
researchers.“

In this chapter. we present a model that belongs to the first tradition.

 

2 This line of literature is nicely surveyed by Gertel (1988). Mishkin (1994) is a
good overview from a global perspective.

3 The basic argument of these studies is that the economy is running an
unsustainable economic program, so that crises are bound to happen. The origin of
this literature is Krugman (1979). who pioneered a study of currency crisis. Since
then, there have appeared countless papers on this topic. Calvo (1995) proposes
several theoretical extensions for modeling capital-market crises. See also
Kaminsky and Reinhart (1996) for a survey study on the potential links between
currency and banking crises. Krugman (1997) is a good introduction, written at a
non-technical level.

‘ See, for example, Agénor and Aizenman (1997).

125

The primary purpose is to characterize the vulnerability of financial markets
using the options approach. It is consistent with the previous chapters, in
that the driving forces of the model are asymmetries in behavior and
information of economic agents. We will see that moral-hazard problems in
financial markets can easily be cast in the option framework. In the next
section, we briefly discuss the nature of moral hazard problems in financial
markets, and illustrate how moral hazard leads to overinvestment in risky
assets. Section 3 is the main section, in which we address two main
features of financial markets caused by moral hazard; we first discuss
overborrowing problems in a simple option framework. Then we propose a
dynamic model of investment to show how a firm’s decision to invest is
affected in the presence of moral hazard. The model will highlight the role
of options under uncertainty and irreversibility that makes financial markets
vulnerable to a crisis. In section 4, we briefly discuss the connection
between financial market problems and the financial crisis. We present two
prominent approaches and interpret their implications in our model

framework. Section 5 concludes by suggesting some policy issues.

4.2. Moral Hazard Problems in the Financial Market

4.2.1. Asymmetries, Options, and Risky Investments

Moral hazard in the financial markets refers to a tendency for a
borrower to engage in activities that are risky from the lender's point of

view. The source of moral hazard is asymmetric information between

 

126

borrowers and lenders. A borrower usually has better information about the
expected return and risk concerning an investment project to be undertaken,
than does a lender. Informational asymmetry may be caused by an
insufficient or costly monitoring system on the part of the lender.

Yet asymmetric information alone is not sufficient for a moral hazard
problem to occur. The borrower must have an option to alter his behavior by
using his informational advantage. Options stem primarily from social
contracts or institutional factors. Health insurance, for example, gives the
insured an option to get medical service whose expense is borne largely by
an insurer. In financial markets, a firm with limited liability has a bankruptcy
option against its debts to a bank. The firm earns a greater portion of the
profits, if it does well, whereas the bank gets only a promised fixed
payment. If the firm goes bankrupt, it has no obligation to repay its loans
fully. and the bank incurs a loss from non-performing loans. Then, the firm
has an incentive to invest in a risky project that would give a good profit, if
it goes well. The riskier the project, the higher the return the firm can
expect from it.5

A moral-hazard problem may arise from another dimension of financial
markets. The governments of most countries provide a safety net to protect

both depositors and financial institutions from bad loans. Deposit insurance

 

5 Asymmetric information also creates ex ante an adverse selection problem. In the
presence of imperfect information about borrowers, the best strategy the bank can
do is to offer a fixed interest rate, and select appropriate projects from a pool of
loan applications received. The end result is that firms with low risk will reject
loans offered at the interest rate that is high from their point of view, leaving those
firms with high risk in the loan market. This is obviously a self-selection process
imposed by asymmetric information. See Stiglitz and Weiss (1981) which is, in
turn, based on a lemons problem, first articulated by Akerlof (1970).

 

127

and loan guarantees are two cornerstones. In the U.S., the Federal deposit
insurance system ensures that depositors are compensated for their losses
from bank insolvency. In many other countries, similar protection systems
are established by laws, or by practice.6 In international borrowing and
lending, it is also usually the case that the government authorities or the
central banks of the borrowing countries—or, sometimes, the lending
country's agencies— provide repayment guarantees to foreign creditors as a
last resort to confidence.

Deposit insurance and loan guarantees were originally designed to
maintain confidence in the financial system, by preventing bank insolvency
and panics. Yet they introduced serious side effects as well. Deposit
insurance, for example, provides a bank with a one-sided option, like a firm's
bankruptcy option. When a bank fails, their losses are covered by deposit
insurance. The insured banks are not exposed to the risk they ought to
bear. Together with loose regulation, deposit insurance encourages moral-
hazard behaviors by the banks: the bank owners do not take full
responsibility for any losses from bad loans, while they gain full credit when
loans make a good profit. Therefore, banks are willing to assume more risk
than they would otherwise.

To make matters worse, deposit insurance is, in many cases, an

unlimited guarantee rather than an insurance with a limited liability that is

 

° Despite these security systems, bank failures flourished in many countries during
the 1980s and 19903, when world financial markets experienced increased liquidity
and highly competitive pressure, following a wave of financial liberalization and
globalization. Recent examples before the Asian crisis include Argentina (1994),
Mexico (1992), Finland (1991). and Sweden (1991).

 

128

common in private insurance. The absence of a coinsurance element

aggravates the problem of moral hazard. A fully insured bank needs not take

 

caution against a big loss, and can exploit risky opportunities to a greater
extent than would otherwise be possible. Depositors also need not be
concerned about the safety of their deposits in the event that banks fail.
Thus, they are totally free from monitoring bank operations and
performance, and from evaluating their riskiness. Also, moral-hazard

problems are more serious in already troubled banks. The higher the risk and

 

the greater the stake, the higher the chance that the bank will be relieved
from the trouble at one shot. They might take any chance to seize a lifeline

desperately.

4.2.2. A Simple Illustration

Moral-hazard problems have been extensively discussed in the context of
many bank failures in several countries. A good example is the recent U.S.
experience in the wake of the savings and loan institutions (S&L) fiasco.7
As this episode has shown, moral-hazard problems encourage risky
investments, and may lead to banking insolvency. When banking panics
occur, the bailout of troubled banks is implemented, and a huge amount of
social loss is unavoidable. A great portion of the burden is obliged to be

borne by the taxpayer.

 

7 Kane (1989) is a book-level treatment on this issue. Shoven, et al. (1992)
discuss the effect of the crisis on real interest rates.

129

To illustrate moral-hazard problems in a formal way, let us examine an
investment problem of a bank. Following Krugman (1998). we assume that
the bank is directly involved in undertaking an investment project. Suppose
that there are two investment opportunities; one is riskless and the other is
risky. The riskless investment bears a sure rate of return r,, which is greater
than zero. The risky investment yields n+8, where a has a uniform
distribution, UI-a, a). How does moral hazard affect the rate of return on the
risky investment? Assuming that the bank is fully compensated for a return
less than r, by deposit insurance, it earns r,+Max[a,0] in any state. To put
it differently. the bank earns a sure return r, plus a European call option on
the stochastic part of the return a. As we have discussed in Chapter 3,
Max]s,0] has a uniform distribution censored at 0. It is not difficult to show

that this distribution is Ul0.al.

Now let us compare the expected rate of return from each investment.
after adjusting for the risk it bears. The riskless investment simply yields r,.
If the investor is willing to accept a sure amount of money equal to d for a
unit of standard deviation, the risk-adjusted rate of return from the risky

investment is

r2=r,+p-$o, (1)
where ,u is the mean and is the standard deviation of U10,a). Since p: é—a,

and O'="71382 . the risky investment with deposit insurance earns

130

7 1—
= + —- —- . 2
r2 r, alz 12¢) ()

The rate of return on the risky investment is greater than that on the safe
investment, unless Z > J3 z 1.7, or the bank is extraordinary risk-averse.8

Note that an increase in a—that is, a mean-preserving spread in the
distribution of a— raises r2, which implies that the higher the risk, the more
the bank would prefer risky investments. Our simple model thus
demonstrates that moral-hazard problems provide an incentive to invest in

risky investments.

The insured banks also exhibit excess demand for risky investment. To
see why. consider another simple model. Suppose that the bank engages in
a loan operation using only capital K, and the production function takes a
simple form. OlK)=In(K-l- 1). Let P and P,, be the price of its output and

input (capital), respectively. The bank’s objective is to maximize profit.
7Z= (P+alln(K+ 1l- P,,K, (3)

where a is a random price shock, such that shocks follow a uniform

distribution, e~U(-a, a) as discussed above. From the bank's optimality

condition, we can derive the demand for capital,

 

' Recall from the discussion of the basic CAPM in Chapter 2 (equation 22) that the
risk premium is expressed as mama, where ¢ is the price of market risk, defined as
¢=(r,,,-r)/a,,,, p”, is the correlation of the return on the project with the return on a
market portfolio, and a is the standard deviation of the project's return. Then

47 =¢p,,,, in equation (1). Unless the investor is extremely averse to risk, If is far

less than 1, which implies that so is 3. See footnote 11 below for more
discussion.

 

 

1. (4)

Obviously, the capital demand for the insured bank and the uninsured bank
depends on Elsi, the expected value for 3. Assuming again that the insured
bank is compensated for any negative shock, then Elal=0 for the uninsured
bank, and Elsl>0 for the insured bank. The moral-hazard problems lead to

high demand for investment.

4.3. Bankruptcy Option, Loan Guarantees, and the Optimal
Investment Rule

4.3.1. The Overborrowing Problem

The foregoing argument assumes that banks engage in asset
acquisition directly. But, in many cases, they make loans to firms to finance
their needs for investments. Therefore, we can examine moral-hazard
problems in financial markets from the perspective of firms that have both a
bankruptcy option and loan guarantees. The moral-hazard view suggests
that the firm tends to depend largely on debts in financing its investments,
and increased borrowing has the undesirable consequence that financial
markets become more volatile and fragile.

To show this formally, let X be the current value of the firm’s asset,
and F be the current value of debt to a bank. Consider now how the asset

value is distributed between stockholders and debtholders of the firm, if the

 

”'1

132

firm goes bankrupt, and is liquidated now. The stockholders who hold
limited liability will receive X-F, if X is sufficiently large, while they will
either gain or lose nothing, if X falls short of F. The payoff to stockholders,

therefore, is
S=Max[X-F, 0]. (5)
The bondholders have a claim equal to
B=Min[X. F]. (6)

which says that they receive either X or F, whicheveris smaller. Assuming
that the manager of the firm represents the interest of stockholders, he or
she will attempt to maximize the equity value, expressed in equation (5).
The claim of stockholders, S, is a call option on the firm’s asset with the
exercise price equal to F. Option theory tells us that the value of a call
option is increasing in uncertainty. When we look ahead. the equity value
will go up as the future value of X is more volatile. The manager of the firm
will, therefore, tend to prefer risky investments to enhance its equity value.
Let us introduce another option (namely, a loan guarantee) into the
analysis. The value of a loan guarantee is a put option on the firm's asset
value, X, with an exercise value equal to the value of the debt, F. It can be

written as
G =Mian-X. 0]. (7)

If X is sufficiently large to exceed F due to a good profit performance, the
option is worthless. On the contrary, if X falls short of F because of a bad

profit performance, the firm will receive a compensation, F-X. The value of

 

"L

133

debt with a repayment guarantee, therefore. is the sum of two values B and

G.

B, = B + G = MI'nIX, F] + MinIF-X, 0] = F. (8)

In other words, the firm has the equivalent of safe debt, since the loan has
no chance of default.

Let us now compare the value of the firms with and without a loan
guarantee. The value of the firm is all outstanding claims to the liquidating

value of its assets. Hence, the firm without a loan guarantee has the value,
V=S+B=Max[X-F, 0]+Min[X, F] =X. (9)

The value of the ordinary firm always equals X, and is not affected by the
value of debt, F. This is the famous result known as the Modigliani-Miller
(MM) irrelevance proposition that the value of a firm is independent of its

financial structure. With a loan guarantee, the firm's value is
V,=S+B,=Max[X-F, 0]+F=Max[X, F]. (10)

Note that the availability of a loan guarantee invalidates the MM proposition
for the beneficiary firm.

Let us now investigate how the value of the beneficiary firm, V,,
responds to changes in the current asset value, X, and the amount of
borrowing with a repayment guarantee, F. It is easy to start with a world of
certainty, where the firm is liquidated now. From equation (10), the value of

the firm is

V, = MaXIX, F].

 

 

134

The firm has a value equal to either X or F, whichever is greater. This
maximum function can be depicted in Figure 4.1. The kinked solid lines
trace two values of V, for F=F0 and F=F, (F,>Fo). respectively. Note that
the amount of borrowing, F0 and F,, constitutes the lower boundary of the
firm value.

We proceed to a world of uncertainty by assuming that the firm is

liquidated in the future. Let X denote the uncertain future asset value that
is stochastic conditional on X. By imposing the expectations operator on
equation (10), and ignoring discounting, the current value of the firm can be

expressed as
v, = E{Max[X -F. 01 + F} = E{Max[X -F. 01} + F: E{Max[X, Fl}. (1 1 I

Let us examine how V, behaves in response to X. When X=O, X
would never exceed F, since the current asset value of zero means that
there is no likelihood that the asset has any value in the future. To
rephrase, if the asset were to have any value in the future, it should be

worth something today. This implies that V,=F at X=O. When X>0, there
is a probability that X is greater than F, so V,>F. The higher the current

asset value, X, the higher probability that its future value, X, is much

greater than F. Then V, becomes far greater than F. If X becomes

extremely large, V, converges to X, since it would almost be certain thatX

is greater than F. Figure 4.2 describes these situations.

 

 

135

 

 

VS
F1
F 0 —-—./
,1’ ’450

 

 

 

Figure 4.1 - Asset Value (X) and the Value of the Firm (V)
(When X is certain)

136

 

 

 

 

 

Figure 4.2 - Asset Value (X) and the Value of the Firm (V)
(When X is uncertain)

137

The solid line traces the path of V, under certainty, while the dashed lines
trace the paths under uncertainty. The dashed line starts with F at X =0.
diverges gradually from the solid line, and finally becomes parallel to that

line at the higher values of X.

Note that the term E{Max[X—F, 0]} in (11) is the call option on the
future asset value. The stockholders hold this option as a claim for their
equity. As we have discussed above, the value of an option is positively
correlated with the degree of uncertainty. Hence. V, exhibits higher values
for any level of X, as uncertainty increases. The upper dashed line in the
figure describes this situation. Finally, we reproduce all the graphs of V,
discussed thus far in Figure 4.3 for the purpose of comparison.

Our discussion offers an important implication in the context of the
firm's risk bearing through the financing policy. For example, consider a firm
that engages in high-risk investments. If it is an ordinary firm, the risk
embedded in the investment raises the possibility of default on debt, so that
debt becomes risky. The debtholders will then require a higher rate of return
to offset this risk-bearing. But if the firm has a safe debt backed by a loan
guarantee. we have a different story. The firm's investment risk can now be
transferred to the government by swapping of risk through borrowing and
with a corresponding loan guarantee. At worst, the government is likely to
fall into a trap, where the firm grows so big as to have a large stake and risk

to be abandoned.

in

 

 

 

 

138

 

 

 

certainty

- uncertainty

 

 

 

 

 

 

 

Figure 4.3 - Asset Value (X) and the Value of the Firm (V)

139

This situation is called the too-big-too-fail problem, which might have a

substantial bearing on many financial crises.9

4.3.2. The Dynamic Model of Investment Decisions

We return to a dynamic model of investment, emphasizing the
Interaction of uncertainty and irreversibility in investment. The model is
consistent with those in previous chapters, with a modification to introduce
the options with regard to bankruptcy and loan guarantee. We examine how
firms make investment decisions in the presence of the options, and how
their decisions render financial markets fragile and inefficient, through moral-
hazard problems caused by the options.

A firm to be considered has the same character that we have assumed
throughout except that it has two options: the (call) option to go bankrupt,
and the (put) option to benefit from the loan guarantee. The investment
project produces one unit of a product over time with an operating cost, C.
The firm borrows 0 (D50), from a bank to cover a portion of its operating
cost. We assume that the loan is interest-free, and its repayment is
guaranteed by the government. If the operating profit, P-C, is less than the

loan repayment, D, the firm declares a temporary bankruptcy. Then the

 

’ The present model is too simple to describe the debt levels of firms observed in
the real world. A lot of studies in the corporate finance literature have introduced
both tax advantages and financial distress caused by borrowing to provide
explanations for observed corporate financial behavior. See Kale and Noe (1992)
for a theory of optimal corporate capital structure. Even in these models, however,
the basic argument for an overborrowing problem under loan-guarantee programs
would remain intact.

140

government intervenes, and bails out the firm under the loan-guarantee
program.10 As mentioned above, the value of the firm consists of three
sources of cash flows: the equity value, the debt value, and the value of the

loan guarantee. Therefore. at any instant, the profit flow is given by
7rlPI=Max[P-C-D. 0]+Min[P—C. D]+Max[D-(P-CI. 0]
=Max[P-C-D, 0]+D
=Male-C. D]. (12)

This equation implies that, if the operating profit were less than the
guaranteed loan (P-C<D), the firm would declare bankruptcy, and receive
full compensation from the government for a default on the loan. Otherwise,
the firm would continue or resume its operation, to earn (P-C). As is the
same as before, we assume a stochastic process for the output price, P,

that evolves according to the geometric Brownian motion.
dP=ant-I-0'Pdw. (13)
The Bellman equationfor the firm’s profit maximization is then given by
VIP) = zlP/dt+ e""'E[V/P+ dP)]. (14)

Using ltO’s Lemma and the limiting arguments as before, we get a

differential equation.

 

1° We assume for simplicity that the firm borrows at no interest, and that the
bankruptcy is temporal. By the latter assumption, we also imply that the firm can
resume its operation without any additional cost. Note, however, that the initial
investment expenditure is yet irrecoverable, if the firm is to exit from industry
forever. These assumptions could be relaxed without changing the fundamental
character of the results.

 

 

h

141

i 02/32 V'IP) + (r-6IPV’IPI-erPl + rrlP) = 0. (15)

The fundamental quadratic equation for this differential equation is
l
3 o’fllfl- 1) + (r-6/fl-r= O, (16)

whose two roots are given by

 

fl, = é-(r-é’l/o’+ Jlrr 41/02 £12 +2r/0'2 >1

 

fl2= %-(r-6)/o’-\/[lr -5//0'2 —§]2 +2r/az <0. (17)

It is straightforward to solve equation (15) by the techniques used in
previous chapters. We have only to distinguish two solutions by the firm's
operating status. The solution for an operating firm, or equivalently when

P-C>D is
VIPI=A,P’3' +7421”: +P/6—C/r. (18)

The corresponding solution is for a bankrupted firm, or equivalently when

P-C<D is
VIPI=B,P" +132P’32 +D/r. (19)

The complementary functions of the solutions represent the value of
options. The term A,P" +A2P’8’ in equation (18) is the value of the option
to go bankrupt, when the price moves into the region (P-C) <D. This option
value goes to zero when P goes to co, so A, should be zero since ,6,> 1. The

term B, P" +82 P’32 in equation (19) is associated with the option to resume

the operation. The option is exercised when P moves into the region

’ I.

 

142

(P-C)>D. This option value has a zero value as P goes to zero, so 8, should
be zero, since ,B,<0. We are left with two parametric constants, A, and B,
that remain to be determined. Noting that VIP) should undergo a smooth
transitional change at P=C+D, we can impose two boundary conditions

with a substitution of C+D for P to get

A,(C+D)’3’ + IC+D)/6-C/r=B,(C+D)A +D/r (value-matching condition).

,62)¢l2(C+D)”*'1 + 1/6= ,6,B, (C +D)’5"1 (smooth-pasting condition). (20)

Solving for A2 and 8, yields

 

= Ir + ,6,5)IC + D) — rfl,C

A .
2 (fl, — mm: + of: r5

 

(21)
(,6, — 132/(c + 0)“ r6

81

Substituting the above into equations (18) and (19) gives a general solution
for VIP).

In Figures 4.1 through 4.3, we have already discussed at a qualitative
level how the value of the firm, VIP). responds to uncertainty and borrowing.
We now present corresponding quantitative results for comparison. To do
this, the base parameter values are set at r=6=0.04 and C=10, and a few
graphs are produced in Figure 4.4. Notice that their shapes are very similar
to those in Figure 4.3; the value of the firm (V) becomes larger, the greater
the uncertainty (6) and the larger the borrowing (D). The only difference is
that we add a little complexity to cut through to other important aspects of

reality to be discussed below.

 

 

143

V0”)
500:
E o=QLD=4
400:

3003 a=atD=4

 

 

 

200* (7:0
100: “ a=0.2,D=2
0’ ,2, f f- ,,,f ,, - .
5 10 15 20 25 30

Figure 4.4 - Output Price (P) and Value of the firm (V)

 

144

Let us now investigate the optimal rule of investment. We consider
two polar cases; one is for completely reversible investment, and the other
is for completely irreversible investment. As we will see ahead, the firm’s
optimal decision rule for investment is affected very differently between
these two extremes. The rule for the reversible investment is easy, and
follows the standard Jorgensonian criterion, which requires that the net
present value (NPV) of a project be greater than zero. If we let K denote the
initial sunk costs associated with the investment, the rule stipulates that
VIPI-KZO. When the investment is irreversible, we ought to consider the
option to invest. The value of the option to invest is the expected future
gains stemming from a possible rise of the output price. Solving for the
complementary function of equation (15). and using the limiting arguments

as before, it can be written as
FIP)=K,P". (22)

' The two boundary conditions again require that VIPI—K should equal FIP) for
their primitive and derivative values at P‘. It is optimal for the firm to
invest, only if PZP", which implies that VIPl-KZFIP). It is also obvious that
the investment will be undertaken, only when the prospect for profitability is
good; in other words. there is no reason to proceed with the investment only
to go bankrupt. It follows that we need to consider the boundary conditions
in the range PZC-I-D only. We can verify this assertion by checking whether
the boundary conditions are satisfied in the region, P<C+D, or not. Thus,

the required conditions for the optimal investment rule are

 

 

145
K,IP*)"” =A2IP*)”2 +P*/6-C/r-K (value-matching condition)

,6,K./P*)f"=,6,A,rP*)"="’+1/6 (smooth-pasting condition). (23)

These equations can be solved for K, and P*.

Let us investigate how P* changes in response to changes in
uncertainty (0') and borrowing (D). At this juncture, it is useful to discuss
the attitudes of firms toward risk. The firm's risk attitude is determined in
various contexts. Firms would be greatly concerned about risk, if the
managers are conservative in selecting investment projects. As another
example, banks would be cautious not to undertake high-risk lendings, when
the government is stringent and effective in supervising bank operations.
Thus, moral-hazard problems may be mitigated by the risk attitudes of firms
and banks.

In our model, the firm's attitude toward risk is reflected in 6. To see

this, recall from our discussion in Chapter 2 that

6=r+¢appm -a, (24)

where r is the riskless interest rate; ¢ is the price of market risk; a is the

standard deviation of P; ppm is the correlation of P with the market portfolio;

and a is the drift rate of P. The second term on the right-hand side
represents a risk premium. When we assume risk-neutrality and ignore the
drift term. the second and third terms will equal zero, so that 6=r. This is
the very assumption we have employed so far. If instead we allow risk-
aversion, 5 alters in response to changes in a: We discuss both the risk-

neutral case and the risk-averse case. For a risk-neutral firm, we assume as

146

before that 6=r=0.04 and a=0. For a risk-averse firm, we assume that

r=0.04, ¢=0.1, pm, =1, and a=0." The initial investment expenditure is

set at K = 300.

Tables 4.1 and 4.2 list the optimal threshold prices, P‘, that are
necessary to trigger investment for risk-neutral and risk-averse firms,
respectively. Not surprisingly, the cost of capital is higher for risk-averse
firms and for irreversible investment. It would be interesting to compare the
effect of risk preference with that of irreversibility. With D=0 and 0' =0.2,
P* is 20.3 for risk-neutral firms under reversibility. If we add risk-aversion,
P* rises to 29.9. If we add irreversibility. P‘ further rises to 47.7. If we
take other parameter values. the irreversibility effect is always stronger than
the risk preference effect. This implies that the option premium dominates
the risk premium in discouraging investment in our parameter setting.12

Let us next examine the effects of borrowing. We have shown that an
increase in borrowing raises the value of the firm, which will, in turn, lower
the cost of capital. In our tables, P' falls as D increases for both

reversibility and irreversibility.

 

“ The term apmin equation (24) is the market (systematic) risk of the project.
Therefore, by setting pp, =1, we assume for simplicity that the systematic risk

equals a. The price of market risk is defined as ¢=Ir,,,-r)/o,,,, where r,,, and 0,, is the
expected return and the standard deviation of the market portfolio, respectively.
Bernheim and Shoven (1989) report that ¢ ranges from -0.03 to 0.45 per year from
1980 through 1988 in the U.S.

‘2 The associated figures may not be directly comparable. since we consider a
particular degree of risk aversion, and complete irreversibility. Still, as we have
discussed before, the results of our model may remain intact in emphasizing the
role of irreversibility, ignored by the neoclassical approach.

147

Table 4.1 - Borrowing (D) and Threshold Price for Investment (P‘)
(Risk-Neutral Firm)

 

Reversible Investment Irreversible Investment

 

 

 

 

Source: Author’s calculations

Table 4.2- Borrowing (D) and Threshold Price for Investment (P‘)
(Risk-Averse Firm)

 

Reversible Investment Irreversible Investment

 

 

 

 

Source: Author 5 calculatIons

 

148

For risk-neutral firms and a=0.2, P* almost halves from 20.3 to 11.0 under
reversibility. by moving from no borrowing to full borrowing. The
corresponding value drops by about one fourth, from 41.6 to 31.2, under
irreversibility. For risk-averse firms, the borrowing effect is rather moderate
in lowering P*, implying that risk-averse firms have less incentive to invest.

Nevertheless, P* is always greater for the risk—averse firm than the risk-

 

"'5

neutral firm, if we hold borrowing and uncertainty constant. It follows that
increased borrowing encourages investment. regardless of the risk character
of firms and the degree of reversibility in investment; and that this effect is
stronger for risk-neutral firms and reversible investments.

Let us examine the other dimension, namely, the effect of an increase
in uncertainty (0'). The effect on risk-neutral firms depends on which project
they are investing in. If they invest in a reversible project, higher
uncertainty reduces P*. If the investment is irreversible, higher uncertainty

raises P’. Thus, whether increased uncertainty encourages or discourages

 

investment depends on the type of investment. The reason is seen by
observing two facts. First, both the value of the firm, VIP), and the value of
the options to invest, FIP). are increasing in uncertainty. Second, FIP) rises
by more than does VIP). as uncertainty increases. Since FIP) is an additional
cost of investing, the net effect of an increase in uncertainty is to raise P*
for the irreversible investment. Risk-averse firms behave in a rather tricky
way with increased uncertainty. Generally speaking, uncertainty is
positively related to P“, when firms are fairly averse to risk. This is

primarily because 6 is increasing in a, so risk-averse firms discount profit

149

flows more heavily, which results in lowering VIP). and raising P*.
However, as we have seen earlier, an increase in uncertainty raises VIP) via
option effects, which, in turn, reduces P*. When a firm invests in a
reversible project by borrowing heavily, the net effect may be a decrease in
P*. The lower left part of Table 4.2 indicates this possibility.

The numerical results suggest a channel through which a firm’s
investment behavior is biased. Let us discuss this, using as a benchmark
the traditional Marshallian long-run average that equals C+rK=22.0 (Recall
that we set C=10, K: 300, and r=0.04 for the base case). The standard
investment theory tells us that. if P is greater than this value, it is optimal to
proceed with the project. The optimal thresholds, P", in our figures are for
the most part larger than 22.0. Thus, the modified rule argues that firms
should invest in a more cautious manner than the standard theory indicates.
There is, however, a perverse case: firms are able to reduce the cost of
capital substantially by borrowing heavily, and investing in easily reversible
and risky projects. For example. Table 4.1 shows that risk-neutral firms can
reduce P* to below 22.0 by investing reversible investments with borrowing.

Our arguments really make a good case to explain why a moral-hazard-
prone economy may well be contaminated by investment volatility. Firms
that prefer risk-taking—motivated by moral hazard due partially to loose
regulations of risky business—are susceptible to highly leveraged, reversible,
and risky investments. The cost of capital for this type of investments falls
artificially to a low level. The consequent overinvestment in reversible and

risky capital implies that such an economy as a whole may have a highly

150

volatile investment climate that is vulnerable to a crisis. The implication in
the context of Asian economies is that ill-advised foreign capital inflows and
domestic credit misallocation may have caused many Asian firms and banks
to be engaged in such investments as real estate, office buildings, and

speculative activities that are highly risky, and easily collapsible.

To show more clearly how investment is affected by asymmetric
information. we push our discussion a little further. Suppose that all
uncertainty is completely firm-specific, so that any risk from uncertainty can
be diversified. This is equivalent to saying that there exists no systematic
risk, and firms can behave as if they were risk-neutral. We assume that
firms are identical in the cost structure. but that they differ by their price
shocks embedded in P. This assumption introduces asymmetric information
about the firms' profitability; firms know their own specific shocks in profit,
but the banks and the government cannot observe them. Accordingly.
outsiders cannot distinguish firms by their cost of capital, P‘.

In Figure 4.5 on the (P,0') plane, the cost-of-capital lines are for risk-
neutral firms. The picture is based on the same base parameter values as
above. The solid lines are for no borrowing (D=0). while the dashed lines
are for full borrowing (D= 10). Since firms are risk-neutral, we know from
the foregoing discussion that P* is decreasing in a for reversible investment,
while it is increasing for irreversible investment. Hence, those lines
associated with reversible investment are downward-sloping, whereas those

lines associated with irreversible investment are upward-sloping.

151

 

 

 

 

 

 

P
Irreversible Investment
/
/ /
1L ’
a,
*=
22 \ I
\
\ .
\ ~ Z __ A ReverSIble Investment
0'

Figure 4.5 - Uncertainty (0') and Cost-of-Capital Lines (P*)

152

Also, since borrowing lowers the cost of capital, downward-slaping curves
get steeper, and upward-sloping curves get flatter, with full borrowing. The
horizontal line drawn at P*=22 is a benchmark cost of capital in the
neoclassical world, where there is no option at work.

Since each firm's cost of capital is not known to outsiders, we ought to
pin down the cost-of-capital line that can be used to evaluate a project from
society’s point of view. It could be an average value of the population, if
the distribution of a is public knowledge. Assume, for expositional
purposes, that the benchmark line I “=22) is the level of the cost of capital
socially agreed on.13 Then it is socially acceptable to invest, when the
output price, P, is greater than the benchmark. From the individual firm’s
point of view, however. it is optimal to invest, if P is greater than each of
the cost-of—capital lines. Hence, there occur mismatches between these two
conflicting criteria. For example, when firms do not borrow, area B is a
portion of firms undertaking the reversible investment that is not socially
desirable. On the other hand, area C+D is a portion of firms that do not
undertake the irreversible investment that is socially desirable. These two
areas together constitute socially unacceptable investments caused by
mismatches. If the banks and the government had perfect information, they

would induce firms to invest. only when their profit returns meet the socially

 

‘3 Note that the benchmark line is for a world of certainty with no loan guarantee
and no irreversibility. In this world, no moral-hazard problem arises, because there
is no asymmetry, and thus the options play no role. By using an arbitrary criterion,
we beg a normative question of what is the socially optimal investment. Using
other criteria, however, would not seriously undermine the basic argument, so long
as asymmetric information exists.

153

desuedleveL

When firms borrow, the dashed lines show that socially undesirable
investments expand to A+B for reversible investment, and shrink to C for
irreversible investment. Thus, borrowing brings good news. as well as bad
news; it tends to encourage firms to engage in socially undesirable reversible
investments, but could also induce them to carry out socially desirable
irreversible investments that incur large sunk costs." This observation not
only reinforces the previous argument that borrowing encourages risky,
reversible investment. but also suggests that it may sometimes be socially

desirable to provide a financial incentive for risky, irreversible investment.15

4.4. Financial Market Fragility and Financial Crisis

We have presented a model to explain how asymmetries and moral
hazard in financial markets give rise to financial fragility that can be blamed
for the financial crises. However, the connection between financial market
problems and financial crisis is not all clear in two respects. First, the
financial and currency crises have also occurred in economies that have
maintained sound financial systems. Most recent examples include Finland

and Sweden; see footnote 1. Second, as mentioned earlier, the economic

 

“ Mankiw (1986) proposes a model in other context to justify the government
intervention in the loan markets. He analyzes the student loan market, in which
information is asymmetric and financing instruments are limited, and asserts that a
public credit program is needed to prevent a collapse, since the unfettered market
equilibrium is unstable.

‘5 Here. we again assume that the social planner wants any project that meets a
guideline to be undertaken. See Emery and McKenzie (1996) for a similar argument
for the government subsidy to large-scale irreversible investments.

154

performances and policy stances of the Asian economies prior to the crisis
are very different from those of other developing countries that experienced
similar crises in the past. It follows that the depth and the scope of the
current crisis has gone farther far than is justified by the fundamentals. This
implies that traditional models of the financial crisis that emphasize the
inconsistent macroeconomic policies may not be appropriate to explain the
Asian crisis; see footnote 3.

There are two lines of modeling efforts to overcome this difficulty. One
approach is based on a classic model of bank runs by Diamond and Dybvig
(1983). In a series of papers, Sachs and his coworkers (e.g., Sachs, et al.
(1996). Radelet and Sachs (1998)) propose a sophisticated version of the
pure financial panic model, which they argue describes the essence of the
recent financial crises in Mexico and the Asian countries. They point to the
fact that the modern high-tech banking sectors are very susceptible to a
panic. A financial panic is a case of multiple equilibria in the financial
markets. A panic is a bad equilibrium outcome in which creditors suddenly
withdraw their loans from solvent borrowers. When short-term debts exceed
short-term assets, and no single private-market creditor is willing to supply
all the credits necessary to pay off existing short-term debts, it is rational
for each creditor to withdraw its credits, if other creditors do so. Then the
crisis becomes self-fulfilling.

Associated with this argument is the investors' sentiment. As long as
euphoria pervades, investors do not care much about risk. When euphoria
collapses. they panic and herd together, only to precipitate the crisis. The

basic message of this theory is that the Asian crisis has little to do with

155

economic fundamentals; rather, it is caused by unnecessary investors' panic.
the absence of international lenders of last resort, and the lack of an orderly
mechanism to deal with international bankruptcy. Not surprisingly, they
criticize the role currently played by the International Monetary Fund (IMF);
they contend that the IMF should act as a lender of last resort, rather than
as a precarious interventionist by imposing strict conditions on the countries'
economic policies and institutions.

The other approach, which is related to the present model, is
associated with Krugman (1998). He takes a different position by
emphasizing moral hazard, asset bubbles. and asset deflation. although he
admits a panic element of the crisis. The key argument of the moral-hazard
story is almost the same that we have discussed above; investors in the
private sector make bad investment decisions that depend heavily on
excessive borrowing and risk-taking, backed by implicit government
guarantees and loose regulations. Krugman contends that the wide-spread
moral-hazard problem deeply rooted in the Asian financial sectors is a
systemic flaw. and thus should be viewed as the mirror image of the
financial panic. He further argues that moral hazard results in overpricing
and vulnerability in the assets markets, and that Asian financial markets,
which attracted a huge amount of foreign capital well into the 19903, were
inflated by an asset bubble. Once the bubble burst, the process abruptly
reversed; plummeting asset prices cripple financial intermediaries, which in
turn triggers further drops in asset prices. In this approach, therefore, an
imbalance in economic fundamentals is not a necessary condition for the

crisis.

156

How can we relate the present model to these crisis theories? We find
an answer by exploring a firm’s disinvestment problem in our model. More
specifically, we examine how firms’ disinvestment decisions are affected,
after investments are undertaken. Suppose that investors lose confidence in
an economy that has a fragile financial system. By the loss of confidence,
we mean that, at a certain point of time as a financial panic is visible, what
seems to have been profitable in good times suddenly looks less profitable or
uncertain to all participants in the market. Then there occur two
impediments to disinvesting. First, investors will have to pay a risk premium
in selling their assets, if uncertainty inherent in the panic is associated with
market risk. Second, all investors will see a probability of collapsing of the
asset markets. which implies that they will incur more sunk cost, the later
they leave out of the markets. In other words, as the financial panic
approaches, irreversibility is created or added in the asset markets, making
the firms’ disinvesting more expensive. In the framework of our model, the
cost of disinvesting may abruptly increase. Hence follow the oversupply of
assets and asset deflation, probably precipitated by self-fulfilling
expectations suggested by panic theory.

Another dominant aspect of any crisis is a clash between the real
sector and the financial sector. When the financial crisis is underway,
interest rates should go up, or the value of currency will drop suddenly,
resulting in rapid capital flight and the fire sale of domestic assets. The key
factors in this process are a sluggish adjustment in the real sector, and

irrational herding and rapid contagion —caused primarily by market

157

sentiments and misinformation—in the financial sector. For example, in his
discussion on the episode of U.S. dollar depreciation in the late 1980s,
Krugman (1989) argues that the slow response of trade balance to the

exchange rate, the so-called J-curve effect, precipitated the currency drop.

4.5. Summary and Conclusions

The Asian crisis seems to have revived an emphasis of moral-hazard
problems in the financial market. The present model points out a firm's
overborrowing problem due to moral hazard in the Modigliani-Miller
framework. If the firm borrows. backed by a loan guarantee. the debt has
no probability of default. Then more borrowing is translated into the higher
firm value. The model shows that a moral-hazard-prone economy tends to
be contaminated by fragile investments. Motivated by explicit or implicit
government insurance, firms are susceptible to highly leveraged. reversible,
and risky investments. In effect, the cost of capital for this type of
investments falls artificially to a low level, which induces overinvestment
that is vulnerable to a crisis. We also argue the effect of asymmetric
information on investment from the society's point of view; that is, if the
firms’ risk is unobservable to outsiders, borrowing tends to encourage firms
to engage both in socially undesirable reversible investments and in socially
desirable irreversible investments with large sunk costs.

What can be done to prevent moral-hazard problems, and to deal with
the legacies they left behind in financial markets? If we are to learn a

lesson from history, there is no best answer. The S&L crisis in the U.S. in

158

the late 19805, and the bursting of the Japanese bubble economy in the
early 19903 demonstrate that the clean-up of the financial sector is usually
protracted and painful. Financial crisis theories differ in their prescriptions.
Roughly speaking, panic theory suggests a gradualism, while moral-hazard
theory recommends a cold-turkey policy; panic theory focus on the
restoration of the investors' confidence. and the enlarged role of
international institutions and foreign banks in providing liquidity to troubled
countries. On the other hand, moral—hazard theory emphasizes the
importance of a comprehensive reform of the bad financial systems that
leads to transparency and efficiency of the market. Clash theory suggests
that transparency should be enhanced. and capital flows should be properly
controlled and regulated.

Some of these policy implications are consistent with those from the
present model. Our analysis suggests that unbounded government loan
guarantees and the risk-taking behaviors of firms, combined with uncertainty
and easy reversibility of investment, deviate the cost of capital from the
socially desirable level. This implies that policy responses to address these
problems include the rule-based government credit programs, an effective
corporate governance and regulatory system, the inducement of diversified
investment portfolios, including long-term investments, and the stabilization
of the price system.

Finally, in a perfect world, where we are able to neatly identify the
sources of the crisis, and there is an omnipotent international lender of last
resort, the prevention of and the recovery from the crises would be much

easier. In the real world. a typical financial crisis is a mixture of various

159

factors, having some moral-hazard elements and some panic components.
Also, the world financial institutions are still not capable of acting as
charitable lenders without intervening excessively in the domestic economic
affairs of the recipient countries. It follows that, if the Asian economies are
to regain their past strength, they would have to restore confidence on their
own. Needless to say, this task must start with correctly addressing

problems and weaknesses in their financial systems.

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160

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