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AN STA

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3 1293 01

 

LIBRARY

Michigan State
University

 

 

 

This is to certify that the

dissertation entitled

CAPITAL INVESTMENT BY RISK NEUTRAL AGENTS:

MERGING ADJUSTMENT COSTS AND IRREVERSIBILITY

presented by

Hirokatsu Asano

has been accepted towards fulfillment
of the requirements for

Ph.D. Economics

degree in

 

 

 

Date May 10, 1999

 

MSU is an Affirmatiw Action/Equal Opportunity Institution

0—12771

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—v‘.

. ._‘ r

-.._ —-r——- v" v fir ‘-— ' v "

PLACE IN RETURN BOX to remove this checkout from your record.
To AVOID FINE return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE

DATE DUE

DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1198 c/CIRCJDmDmpGS-p.“

CAPITAL INVESTMENT BY RISK NEUTRAL AGENTS:
MERGING ADJUSTMENT COSTS AND IRREVERSIBILITY

By

Hirokatsu Asano

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Economics

1999

Current <
firm mal
capital i1
Optimal
anal} 3i3
include3
Shows U
the disc

anal} Sis

ABSTRACT

CAPITAL INVESTMENT BY RISK NEUTRAL AGENTS:
MERGING ADJUSTMENT COSTS AND IRREVERSIBILITY

By

Hirokatsu Asano

Current capital investment affects future investment by setting conditions upon which a
firm makes future investment decisions. This analysis applies option pricing theory to
capital investment in order to determine possibilities of future investment, and shows the
optimal investment choice for a firm contemplating future investment. The theoretical
analysis develops a model for capital investment by risk neutral agents. The model
includes costly reversibility and fixed costs of investment. Then, numerical analysis
shows that optimal investment is approximately linear in economic parameters such as
the discount rate. Empirical analysis shows that actual investment behaves as theoretical

analysis predicts.

 

\Vi
been com;
he gate m;
and his CD!
the Other H
010m. it
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I m
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and 3111(1)
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grateful to

achicVemr

ACKNOWLEDGMENTS

Without the help and support from many people, this dissertation could not have
been completed. First, I would like to thank Professor Robert Rasche for guidance that
he gave me throughout this work. An article that he suggested me initiated this research,
and his comments and suggestions improved this work greatly. I would also like to thank
the other members of my committee, Professors Jeffery Wooldridge and Gerhard
Glomm, for their insight and support. I am truly thankful to all members of my
committee for the quality and the speed of their feedback on my work.

I am very fortunate to have good friends at Michigan State University. I
especially want to thank Daiji, Heather, Hiroki, Katsushi and Pablo. They made my life
and study in Lansing enjoyable, if not pleasant, and some of them also contributed to my
work.

Finally, I thank my parents for allowing me to pursue my Ph.D. dream. I am also
grateful to my brother and his family for their support. They made my academic

achievement possible.

LIST 0

LIST 0

INTRO

CHAP
0an

b)
I
I.)

TABLES OF CONTENTS

LIST OF TABLES ............................... vii

LIST OF FIGURES ............................................................................................... ix

INTRODUCTION ................................................................................................... 1
1. Irreversibility, Costly Reversibility and Fixed Costs of Investment ............. 2
2. Prior Work .................................................................................................... 4

CHAPTER 1

COSTLY REVERSIBLE INVESTMENT WITHOUT FIXED COSTS ................ 8
1—1. Bellman Equation ...................................................................................... 9

1-2. Optimal Investment for Costly Reversible Investment

without Fixed Costs ............................................................................. 11
CHAPTER 2
COSTLY REVERSIBLE INVESTMENT WITH FIXED COSTS ....................... 18
2-1. Optimal Investment for Costly Reversible Investment
with Fixed Costs .................................................................................. 19
2-2. User Cost of Capital ................................................................................ 23
CHAPTER 3
OPTIMAL INVESTMENT AND EFFECTS OF PARAMETERS ...................... 25
3-1. Optimal Investment Rule ......................................................................... 25
3-2. Effects of Parameters on Optimal Investment ......................................... 27

iv

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CHAP'
ECOXI

44.

$3.

CHAPT
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54.

fit

54-8

CHAPTER 4

ECONOMETRIC PROCEDURE .......................................................................... 44
4-1. Econometric Model ................................................................................. 44
4-2. Analysis Methods .................................................................................... 47

i. Data Sources .......................................................................................... 47
ii. Procedure .............................................................................................. 48

CHAPTER 5

INDUSTRY ANALYSIS ....................................................................................... 52
5-1. Computer and Office Equipment Industry .............................................. 52

i. Estimation and Measure of Zero Investment ......................................... 52

ii. Test of Model Selection ........................................................................ 57

iii. Test of Serial Correlation .................................................................... 58
5-2. Automobile Industry ................................................................................ 58
i. Estimation and Measure of Zero Investment ......................................... 58

ii. Test of Model Selection ........................................................................ 63

iii. Test of Serial Correlation .................................................................... 63
5-3. Airline Industry ........................................................................................ 64
i. Estimation and Measure of Zero Investment ......................................... 64

ii. Test of Model Selection ........................................................................ 71

iii. Test of Serial Correlation .................................................................... 71
5-4. Summary .................................................................................................. 72
CONCLUSIONS .................................................................................................... 74

 

 

APPEXD
IXVE

A-l.

APPEXDI
AND E

APPENDIX A. TWO PERIOD MODEL OF COSTLY REVERSIBLE

INVESTMENT WITHOUT FIXED COSTS .................................................. 76
A-l. Investment Model ................................................................................... 77
A-2. Values of Future Investment and Disinvestment.... ................................ 79
A-3. Optimal Investment for First Period ....................................................... 80
APPENDIX B. APPROXIMATION OF G SATISFYING J(R,G) = O ................ 85
B-I. Approximation by Order of Exponents ................................................... 85
B-2. Approximation by Binomial Series ........................................................ 86
B-3. Refinement of Approximation ................................................................ 87

APPENDIX C. FUNCTION q WITH AND WITHOUT FIXED COSTS

AND RANGE OF INACTION ........................................................................ 89
C-1. Coefficients of Function q, 8,. and 8,. ..................................................... 89
C-2. Range of Inaction .................................................................................... 90
APPENDIX D. SERIALLY CORRELATED ERROR ........................................ 91
D-l. Voung’s Test of Model Selection ........................................................... 91
D-2. Estimated p and Durbin-Watson Test of OLS Residuals ....................... 94
D-3. Discussion ............................................................................................... 95

APPENDIX E. TWO STEP ESTIMATION FOR
LIMITED DEPENDENT VARIABLE MODEL ............................................ 96

BIBLIOGRAPHY ................................................................................................ 100

vi

 

 

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Table 3.8 In

LIST OF TABLES

Table 1 Recent Literature on Adjustment Cost Function and Irreversibility ..................... 5
Table 3.1 Cases for Simulations ...................................................................................... 28
Table 3.2 Significance of Discount Rate ......................................................................... 32
Table 3.3 Critical Value for Investment without Fixed Costs, yi .................................. 33
Table 3.4 Critical Value for Disinvestment without Fixed Costs, y' ............................. 34
Table 3.5 Target Value for Investment with Fixed Costs, yL, ........................................ 35

Table 3.6 Trigger Value for Investment with Fixed Costs (Natural Logged), Ln y; ..... 36

Table 3.7 Target Value for Disinvestment with Fixed Costs, yi-u ................................... 37

Table 3.8 Trigger Value for Disinvestment with Fixed Costs, y}, ................................. 38
Table 3.9 Significance of Quadratic Terms and Interaction Terms ................................. 40
Table 3.10 Measure of Minimum Investment (Natural Logged), Ln G” ...................... 41
Table 3.11 Measure of Minimum Disinvestment (Natural Logged), Ln G‘ .................. 42

Table 3.12 Measure of Inaction Range (without Fixed Costs, Natural Logged), Ln G..43

Table 4.1 Investment Models and Econometric Methods ............................................... 49

vii

 

 

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Table-33 1351
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Table 5.12 Ser

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Table 5.1 Measure of Zero Investment (Computer and Office Equipment Industry) ...... 53

Table 5.2 Estimation of Computer and Office Equipment Industry ................................ 56
Table 5.3 Model Selection for Computer and Office Equipment Industry ...................... 57
Table 5.4 Serial Correlation Test for Computer and Office Equipment Industry ........... 58
Table 5.5 Measure of Zero Investment (Automobile Industry) ....................................... 59
Table 5.6 Estimation of Automobile Industry ................................................................. 62
Table 5.7 Model Selection for Automobile Industry ....................................................... 63
Table 5.8 Serial Correlation Test for Automobile Industry ............................................. 64
Table 5.9 Measure of Zero Investment (Airline Industry)............................ ................... 65
Table 5.10 Estimation of Airline Industry ....................................................................... 70
Table 5.11 Model Selection for Airline Industry ............................................................. 71
Table 5.12 Serial Correlation Test for Airline Industry ................................................... 72
Table D.l Log Likelihood of Investment Model ............................................................. 92
Table D2 Model Selection under Serially Correlated Error ........................................... 93
Table D3 Estimated Coefficient of AR(1) Process, p ..................................................... 94
Table D4 Durbin-Watson Statistic for OLS Residuals ................................................... 94

viii

 

 

Figure 1.1

Figure 1.3

Figure 1.3

figure 1.4

HSure 2.1

FiSure 2.2

n,-

(11

O

0

Figure 1.1

Figure 1.2

Figure 1.3

Figure 1.4

Figure 2.1

Figure 2.2

Figure 2.3

LIST OF FIGURES

Approximation of J(G,R) = 0 ......................................................................... 13
Costly Reversible Investment without Fixed Costs ....................................... 14
Optimal Rule for Investment in

Costly Reversible Investment without Fixed Costs Model ....................... 17

Optimal Rule for Disinvestment in
Costly Reversible Investment without Fixed Costs Model ....................... 17

q(y) and N02) for Costly Reversible Investment with Fixed Costs ................. 21

Optimal Rule for Investment in
Costly Reversible Investment with Fixed Costs Model ............................. 22

Optimal Rule for Disinvestment in
Costly Reversible Investment with Fixed Costs Model ............................. 23

Figure 3.1 Schematic Diagram for Costly Reversible Investment with Fixed Costs ...... 25
Figure 3.2 Critical Values for Investment and Disinvestment ......................................... 29
Figure 4.1 Normalized Investment Function ................................................................... 44
Figure A] (102), N02), P'(y), and C' (y) for Two Period Model ....................................... 82
Figure A.2 Optimal Rule for Investment in Two Period Model ...................................... 84
Figure A.3 Optimal Rule for Disinvestment in Two Period Model ................................ 84

ix

 

 

 

Figure 8.1

Figure 8.2

Figure B]

Figure 8.1 Approximation of 6* (I) ............................................................................... 87
Figure B.2 Approximation of G* (2) ............................................................................... 88

Figure D.1 GFM vs OLS .......................................................... ’ ....................................... 95

 

 

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INTRODUCTION

When a firm invests, its investment decision is based upon its current level of capital
stock. Then, firm’s current investment affects its future investment decisions by setting
conditions upon which the firm makes its future decisions. Therefore, the firm should
take into account the possibilities of future investment when it makes its investment
decision. This paper shows the optimal investment decision for a firm contemplating
future investment, and investigates actual investment in accordance with theoretical
analysis.

This analysis applies option pricing theory to capital investment in order to take
into account the possibilities of future investment. Currently, Abel and Eberly are
merging Tobin’s q theory with an adjustment cost function and irreversible investment in
order to incorporate the possibility of future investment. Abel and Eberly (1996) use a
model with costly reversibility but without fixed costs of investment. Abel and Eberly
(1997) use a model with fixed costs of investment and irreversibility. This paper uses a
model with both fixed costs of investment and costly reversibility. In general, a closed
form solution of this model is unattainable. Therefore, this analysis solves the problem
numerically. The theoretical analysis shows that Tobin’s q becomes lower, when a firm
takes into account its future investment.

Tobin’s (marginal) q theory implies that a firm invests when the marginal benefit
of capital exceeds the marginal cost of capital. The firm will invest up to the point where
the marginal benefit of investment becomes equal to the marginal cost. The marginal
benefit of capital is Tobin’s q, and equal to the derivative of the present discounted value
(PDV) of net cash flow or firm’s equity. The marginal cost is the price of capital.

Investment is an increasing function of Tobin’s q.

 

 

 

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This analysis modifies the friction model to analyze actual investment. The
analysis compares five different investment models including one that specifies costly
reversible investment with fixed costs. The analysis shows that costly reversible
investment or the corresponding econometric model, a generalized version of the friction
model (GFM), is the best model to analyze actual investment among the tested models.
The analysis also shows that an investment model of partial specification such as
irreversible investment can be inferior to a model of reversible investment without fixed
costs for empirical research.

1. Irreversibility, Costly Reversibility and Fixed Costs of Investment

If the firm cannot sell its installed capital, investment is irreversible. When the firm
invests heavily now, it is more likely that the firm will have too much capital in the future
and Tobin’s q will be lower than the price of capital. In other words, the marginal benefit
of capital will be less than the marginal cost of capital in the future. In such a case, the
firm will want to sell its installed capital, but cannot due to the irreversibility of capital.
On the other hand, if the firm postpones its investment decision, it will acquire more
information about the economy and, then, can adjust its capital stock. Thus, waiting has
value for the firm in a stochastic economy, and the firm’s gain from waiting is more
accurate adjustment of its capital stock in the future. The irreversible investment
literature applies option pricing theory in order to evaluate the value of waiting.

The irreversibility literature regards future investment as a call option. A call
option is a right, but not an obligation, for investors to buy financial assets or
commodities at a predetermined price, or striking price, at a predetermined date in the
future. The investors or option buyers exercise their option if the market price of the
assets exceeds the striking price at the contract date. If the market price is lower than the
striking price, the option buyers will not exercise their option. The gain for the option
buyer is difference between the striking price and the market price at the maturity date.

Therefore, the gain is kinked at the striking price. The gain is zero if the market price is

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lower than the striking price. If the market price is higher than the striking price, the gain
is positive and increasing along with the market price. The premium that the option
buyers have to pay to option writers is equal to the expected gain for the option buyer.
Option pricing theory evaluates this premium.

In the stochastic economy in this paper, operating profit and cash flows are
modeled as functions of capital stock and a stochastic variable, which is called the
economic indicator variable. For a given level of capital stock, there is a critical value for
investment of the economic indicator variable, which corresponds to the striking price.
The firm invests when the economic indicator is higher than the critical value.
Otherwise, the firm does not invest, because the marginal benefit of investment is less
than the marginal cost of investment.

When the firm can sell its installed capital at a resale price lower than the
purchase price of new capital, investment is costly reversible. Costly reversibility
introduces another critical value of the economic indicator variable, the value at which
disinvestment becomes profitable. The firm disinvests if the economic indicator is lower
than the critical value for disinvestment. Then the firm will disinvest until Tobin’s q
becomes equal to the resale price of capital. The firm does not disinvest if Tobin’s q is
above the resale price. When Tobin’s q is between the purchase price of capital and the
resale price of capital, zero investment, or inaction, is optimal. Similar to the gain from
investment, the gain from disinvestment is kinked at the critical value. The gain is
positive when the economic indicator is below the critical value and zero when the
economic indicator is above the critical value; In the analysis, disinvestment is similar to
a put option.

When there are fixed costs for investment, fixed costs introduce a minimum
amount of optimal investment. Sunk costs or installation costs of investment are
examples of fixed costs of investment. Because of fixed costs, a value of Tobin’s q

higher than the price of capital is not sufficient for optimality. An increase in the PDV of

 

 

 

 

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net cash flow from investment can be smaller than fixed costs. When the firm invests,
the gain from investment should be larger than fixed costs. As a result, optimal
investment has a minimum. If there are fixed costs for disinvestment, optimal
disinvestment also has a minimum. Fixed costs also introduce a range of inaction in the
economic indicator.
2. Prior Work
Abel and Eberly have published several papers about capital investment by risk neutral
agents which incorporate Tobin’s q theory with an adjustment cost function and
irreversibility. Abel and Eberly (1994) formally incorporate fixed costs of investment
into their model. They solve a continuous time model with a general form of the
investment cost function which includes both fixed costs and the costly reversibility of
investment as well as a conventional convex adjustment cost function. Their analysis
shows that a range of inaction appears in the firm’s investment decision. Both fixed costs
and the costly reversibility result in inaction as an optimal investment decision. In one
case, a constant returns to scale (CRTS) Cobb-Douglas technology and a perfectly
competitive market are assumed. This model results in the firm showing risk-loving-like
behavior, i.e., the firm will invest more when the price of output fluctuates more, because
the operating profit function is convex in the output price under those assumptions.
When the firm contemplates an investment project, the firm can wait until the
firm’s market condition becomes more favorable. McDonald and Siegel (1986) employ
option pricing theory to evaluate the value of waiting. The irreversible investment
literature shows that optimal investment has two states; one is strictly positive investment
and the other is zero investment, and one of the two states appears at a time. Bemanke
(1983) by employing option pricing theory shows an asymmetric character to irreversible
investment. When market conditions for the firm are favorable, the firm will invest and
acquire more capital. Because the firm cannot resell its installed capital, the firm should

be worried about “bad news.” Due to the irreversibility of investment, the firm cannot

 

 

 

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sell its installed capital, even if a lower capital stock is optimal. In other words, the firm

can adjust its capital stock upward but not downward. Thus, the firm’s investment

decision is concerned only with bad news, because what “irreversible investment is

sensitive to is ‘downside’ uncertainty (Bemanke, pp. 93).”

Currently, Abel and Eberly are merging the adjustment cost function literature

and the irreversibility literature. Table 1 shows recent articles and the topics which the

articles cover. Abel, Dixit, Eberly and Pindyck (1996) develop a model which

incorporates both Tobin’s q theory and option pricing theory. The model is a two period

model with the costly reversibility, but without fixed costs. One conclusion from that

model is that when the firm makes a decision it should take into account the adjustment

of its capital stock in the future (the second period). The appropriate q for current

investment is the derivative of the PDV of cash flow assuming no future investment or

disinvestment, plus the value of the put option for future disinvestment, minus the value

of the call option for future investment. Their analysis also shows that an upward shift in

the distribution of shocks to the firm’s revenue increases the incentive to invest, while a

mean-preserving spread in the distribution of the shocks has an ambiguous effect on

investment.

Table 1 Recent Literature on Adjustment Cost Function and Irreversibility

 

 

 

Adjustment Cost Function Irreversibility Model
Papers‘ Costly Fixed Convex Irreversi- Option Two Continuous
Reversibility Costs Function bility Pricing Period Time

AE (1994) X X X X
ADEP(1996) X X X

AE (1996) X X X

AE (1997) X X X X
Asano(l 999) X X X X X

 

 

 

 

* AE and ADEP stand for Abel and Eberly, and Abel, Dixit, Eberly and Pindyck,

respectively.

_ ..—— "”5

 

 

 

 

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Abel and Eberly (1996) use an investment model with the costly reversibility, but
without fixed costs. The paper shows that the range of inaction is wider than that
calculated from the Jorgenson’s user cost of capital, i.e., (real interest rate + depreciation
rate) x (purchase price or resale price of capital). Their model uses an operating profit
function which has constant returns to scale in capital stock and a demand shock variable.
The analysis uses a Taylor approximation for its solution, but there are large
approximation errors in some cases.

Abel and Eberly (1997) use a model incorporating both irreversibility and fixed
costs. The model shows that there are a trigger value and a target value for a composite
variable, i.e., a random variable representing the ratio of economic conditions to current
capital stock. When the composite variable exceeds the trigger value, which means that
the economy is booming, the firm increases its capital stock such that the composite
variable becomes the target value. The model uses a Cobb-Douglas technology, and an
iso-elastic demand function. There are stochastic shocks to technology, the demand for a
firm’s output and the price of flow input. The model also includes the level of factor
utilization. .

The analysis presented here uses a modified Abel and Eberly (1997) model with
the costly reversibility. The analysis includes fixed costs, but excludes the factor
utilization. The model without fixed costs is a special case of the model with fixed costs;
namely fixed costs are zero. The analysis will quantitatively show optimal investment
and effects of economic parameters on optimal investment. In order to derive critical
values for optimal investment, there are two obstacles: solving a quotient of power
functions for the model without fixed costs and solving a simultaneous equation system
0f power functions for the model with fixed costs. This analysis resorts to numerical
solutions.

Since costly reversible investment has three ranges in the economic indicator

variable, the analysis employs a generalized version of the friction model. Maddala

 

 

 

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(1983, pp. 162) discusses this model. The friction model is an extension of the Tobit
model, which has two ranges. Tobin (195 8) studies a model in which a dependent
variable is varying in one of the two ranges while it remains constant in the other range.
However, the economic model of this analysis has three ranges in the economic indicator:
one for investment, zero investment and disinvestment. The economic model requires a
modification to allow some explanatory variables to have different coefficients in
different parts of the friction model. Thus, the analysis generalizes the friction model for
analysis.

The analysis presented here compares five investment models: reversible without
fixed costs, irreversible without fixed costs, irreversible with fixed costs, costly reversible
without fixed costs, and costly reversible with fixed costs. Since some examined models
are non-nested, the LR test is not appropriate for comparison. Voung (1989) proposes a
test of model selection, which is an extension of the LR test. The analysis employs
Voung’s test of model selection.

The empirical analysis investigates three industries: the computer and office
equipment industry, the automobile industry, and the airline industry. For all three
industries, costly reversible investment with fixed costs or the corresponding generalized
friction model is the best among the five investment models, since its likelihood is

highest and its estimated coefficients are compatible with the economic model.

 

 

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CHAPTER 1

COSTLY REVERSIBLE INVESTMENT WITHOUT FIXED COSTS

This chapter analyzes costly reversible investment without fixed costs. The analysis

assumes that the operating profit function, 7:, has the following functional form.

7r(K,,Z, ) = A,,Z,""K,” (1)

Here, Z, and K, are the stochastic economic indicator variable and capital stock,
respectively. The specification of equation (1) can be derived for a firm with a CRTS
Cobb-Douglas technology facing an iso-elastic demand curve. The economic indicator
shifts a firm’s operating profits and is assumed to follow a geometric Brownian motion
with drift ,uz and volatility 0'2.

dZ, =,uZZ,dt+0'ZZ,dzz (2)
Here, dz 2 is a standard Wiener process.

The firm invests or disinvests to maximize the firm’s expected equity or the PDV
of firm’s expected net cash flow, V(K,,Z,). The investment and disinvestment incur total
costs, IC,, which include payments for investment, and receipts from disinvestment. The
functions I, and D, are a cumulative investment function and a cumulative disinvestment
function, respectively. The function I, is the sum of all purchases of capital up to time t.
Therefore, d1, is the amount of investment at time 1. Similarly, D, is the sum of all sales
of capital up to time t. Both are non-decreasing step functions. And, because capital

stock, K,_, is exponentially depreciating at an exogenous rate, 5. while the functions I, and

 

 

 

D: remain the

K :1. D V

Here}: [7; 111‘
price ofcapital
the firm makes
reyealed. in orc
assuming optin'
l-l. Bellman I
By splitting the
the second peri 1'

Spread. 7~ Nu. is

[IKI'ZI ) : E,‘

= A». A‘

+ e 7””

s ,4: Z,

The "lain text 01

:Wrmhes a Ivy 0

1 0.110 C I“tuition

bacon] tit and Silt

' 0d 0 '.
el

D, remain the same level until the next investment or disinvestment, in general,

K, at I, — D,. We can write the value function, V(K,,Z,), as follows.l

V(K,,Z,)z max E,l:J:e'”{A,,Z,'j,”K,’i.ds—IC,,,}] (3)

{ditto J11)!” i

subject to dK,,_,. = d],,_. — dD,,,. — 5 K,,,.dt, Km. 2 0 Vs 2 O,
dZHs : [1221+de + 0.2 Zt+stZ a and
ICHs = pKdIH-s _ pKdDHs

Here, y, p; and p; are the discount rate, the purchase price of capital and the resale

price of capital, respectively. Parameters y, p; and pg. are given. At a point in time, t,
the firm makes its decision about K, after the stochastic economic indicator variable, 2,, is
revealed, in order to maximize its expected equity or expected PDV of the net cash flow
assuming optimal investment for the future.

l-l. Bellman Equation

By splitting the time period of equation (3) into two: the first period from zero to AI, and
the second period from At to infinity, and assuming that At is sufficiently small and the

spread. y— ,uz, is strictly positive, we have

V(K, , Z, ) = E, [ f e‘” A, Z,’,‘fK,‘i_,ds] + max E, i: fie” {AnZIl+—.i') Kris-(1S - [Cm i]

I‘ll!“ JII)!1‘.I‘I

Z, >13

+3—7A’ max E’|:J:e-fl {Aan-FzHrKZAHrdT— pKdIHAHr + pKdDHAHr }]

{‘III+AI*fv(ll)I-t-~+f}

= My pe—<r+wt-~'E(z:;s

 

z AKZIFQKIBAI + EI[V(KI+AI9ZI+AI )] (4)

 

1+7At

 

I The main text of this paper solves a continuous time model, while the appendix A of the
paper solves a two period model. Although the two period model of this paper shows
economic intuition, it is the continuous time model which allows us to derive the optimal
investment and show effects of parameter variation on the optimal investment. In the two
period model of this paper, for simplicity, a firm makes investment decisions only twice;
one at the beginning of each period, so that the second period decision does not concern
future investment.

 

 

 

 

Therefore.

By diyiding b

By Ito’s lemn
Thus. the Bell

Becaus
the exl‘t‘nents
diITerential eqt
defining ,I' E Z:

Wind order 0

The SOIUIIOI'] 0f
HIE firSI [em] 0

I 1
he characten' St;

The Second tem

E

Therefore,

rAtV(K.,Z, I = A.Z,""K," (1 + WIN + E.IV(K,.A,,Z,..1, )- V(K,,Z, )I
= A,Z,"9K,” (1 + mm: + E, [AV(K,,Z, )]. (5)

By dividing both sides by At and letting At —> O, the above equation becomes
Ill/(Kr ’ZI ) = Athl-oKrg+(1/d’)Et[dV(K19 21)] (6)

By Ito’s lemma, E,[dV( - )] can be expressed as follows:

E, [dV(K, ,Z, )] = —5 K,VKdt + szzZ,dt +é-szafiZfdt. (7)
Thus, the Bellman equation for V( - ) is

7V(K,,Z, ) = A,Z,""K,” — 5 Ky, + fizzy, + $032,2sz . (8)

Because V( - ) is set up to be homogeneous of degree one in K and Z by choosing
the exponents on Z, we can transform the partial differential equation (8) to an ordinary
differential equation. The derivative of the expected equity, VK, is equal to Tobin’s q. By
defining y a Z, / K,, ,u, a ,uz + 6, and a; 2 0'2, q(y) ( 5 VA) becomes the following
second order ordinary differential equation

2

(7 + 5Iq(y) = A149 y” + #.qu'(y)+ %yzq"(y). (9)

The solution of equation (9) is

q(y) = Hy”) + BNy‘”‘ + B,.y“’". (10)
The first term of equation (10) is the particular solution, and H = AKB / f (1 — 61). Then

the characteristic function f associated with equation (9) is given by

mos—92w -[#_y-32’—']¢+(7+5)- (11)

The second term and the third term of equation (10) are the complementary solution. The

second term corresponds to the derivative of the value of future disinvestment as the put

10

 

 

 

 

option and 11
the call optic
I-Z. Optim:
The optimal
ot‘costly rey:
purchase pric
resale price 0
to the corresp
When
one for intest.
lithe firm 1':
Place. \yhile be

Similar Propert

and

“16 first two cor

Lhe high Order Co

I“ . B.” and 8,,
Abei and 1

hon .

option and the third term corresponds to the derivative of value of future investment as
the call option. And, the exponents (pp (< 0) and (0,. (> I) solve fl (p) = 0.
1-2. Optimal Investment for Costly Reversible Investment without Fixed Costs
The optimal investment decision equates Tobin’s q with the price of capital. In the case
of costly reversible investment, the firm should buy new capital when q exceeds the
purchase price of capital, while it should sell its installed capital when q is lower than the
resale price of capital. Then, the firm invests or disinvests until Tobin’s q becomes equal
to the corresponding price of capital.

When there are no fixed costs for investment, there are two critical values in y:

one for investment and the other for disinvestment. At the critical value for investment,
y+ , the firm is indifferent between investing and waiting. Above y" , investment takes

place, while below y+ the firm waits. The critical value for disinvestment, y” has

similar properties. Then, the boundary conditions for equation (10) are fourfold:

q(y‘)=pZ-, (128)
q(y‘)=p;-, ' (12b)
q'(y*)=0. (12c)
and q'(y‘)=0. (12d)

The first two conditions are the smooth pasting conditions. The last two conditions are
the high order contact. For investment without fixed costs, there are four unknowns, y+ ,
y~ , BN, and B”.

Abel and Eberly (1996) derive the solution to equation (10) which satisfies the
boundary conditions (12a) to (12d) as

Bi, = —(‘—'¢‘-’)—Hg(6)(y‘ )""“"‘ , (13a)
N

3,. =—-(‘—‘—9)—’i[1—g(0)1(y')""“”". (13b)

I)

ll

 

 

 

 

and

Here. U 5 .1.

dosh—U-

equation

ohere. R E 17,-:
Abel an

point (I? = 1 am

But. usually R >
approximation. t1
For example. “he
actually 5.43. An

West term in ei t

he function 17. T Flt

aP'Proximation as fi

appendix B ,

 

. p2 ":5
__ , 13
y iA,ahG"i ' ( C)

and y‘ = [TAT—pL—jT—o. ' (13d)

Here, G a y’ /y-, g(x)E (x"’" —x'—")/(x“’r -—x"’" ) and
h(x) a [1 — (1 — t9)g(x)/(0N — (1 - 6){1 — g(x)}_/¢,, 1/_/'(1 — 0). Here, G satisfies the following
equation
J(R,G) a Rh(G)— G"”h(G" )= 0 (14)
where, R a p; /p,; .
Abel and Eberly (1996) solve equation (14) by a Taylor approximation at the

point (R = l and G = 1). They suggest

60'2

G=1+i(t—9)(;+6

But, usually R > 1 and G > I. When the solution is far from the point of the Taylor

 

)]M (R -1)'”‘ . (15)

approximation, there is a large approximation error. Figure 1.] shows such an example.
For example, when R = 2, Abel and Eberly suggest that G is approximately 1.51, but G is
actually 5.43. An alternative approximation for equation (14) can be derived by choosing
a larger term in either the numerator or the denominator in the function g for simplifying
the function h. The alternative approximation yields equation (16), which is a better

approximation as figure 1.] shows.2

G {[MIKP—jm—mijw (16)
(01' ‘1+6 CON

 

2 See the appendix B for derivation.

 

 

 

 

Equation (16) 5
equation ('14) n

Figure 1
costs. Since 0 «-
function. H1
,Dositiyey. And.
iiunction. 8.j~'

intinite at}: z 0 l

B. < 0. the third t
The third term eq
infinity. So that it

i», ’here are a I Um I

it the q funCllon at

tsdtattn as a fume“

0-351 '1 . _

10'

 

 

 

 

Figure 1.1 Approximation of J(R,G) = 0

(Oz 0.143, A,z 0.506, 5: 0.06, y: 0.07, #2 = 0.05, 0'; = ,uz)

Equation (16) suggests that G is approximately 6.07. However, this analysis solves
equation (14) numerically in order to find more accurate results.

Figure 1.2 shows one solution for costly reversible investment without fixed
costs. Since 0 < t9< 1, H> 0 and O < 1 — t9< 1. Therefore, the first term of the q
function, Hy” , which is equal to the so-called naive case, is concave and increasing for
positive y. And, the term is zero at y = 0. Since (0N < 0 and Bo > 0, the second term of the
q function, 8,, y"’"' , is convex and decreasing for positive y. As the second term is
infinite at y = 0, it dominates the other terms for small positive y. Since (pp > 1 and
B, < 0, the third term of the q function, B,.y"’" , is concave and decreasing for positive y.
The third term equals zero at y = 0, and approaches negative infinity as y approaches
infinity, so that it dominates the other terms for large positive y. As a result, for positive

y, there are a local minimum of the q function at a small value in y and a local maximum

of the q function at a large value in y. Figure 1.2 (a) shows the (marginal) q ratio, which
is drawn as a function of y. The q function is negatively sloped for y < y“ , then

positively sloped, peaks at y” , and again becomes negatively sloped. The q function is

 

 

 

FIQUIL

l9s0l43

 

 

(a) Functions q (y) and My)

 

 

 

 

 

 

q t y )
p x
1——-— . \-—
l
I
l
I
0 5—-/— I
/ |
I
I 1
’ ( D ) y 1' t I >
0
2 Y
(b) Optimal Investment and Disinvestment
d I t
/ ( I )
t 0 > y
0 [/r Z t /K t

 

 

Figure 1.2 Costly Reversible Investment without Fixed Costs

(19:: 0.143, A,z 0.506, 5: 0.06, y: 0.07,;1, = 0.05, a, = 112. G e 5.43)

 

 

tangential [0
function is 11‘
is presented 1
,\' function i5
For Ci

abQVC .1" 01'
inyests or dis:
Value. y" 01' .
optimal. and c
installation of
For the naiye c
left of the costl
than those for t
Figure 1
function of the 1
In figure 1.2 t b)

reyersible inyest
lines are deriyed
myestment is opt

Optimal inyestme

Ifthe ratio is beloy

tangential to the price lines ( p; = 1 and p; = 0.5 ) at the critical values of y. The N
function is the so-called naive case in which the firm assumes zero future investment. It

is presented as a dashed line in figure 1.2 (a). The bold line of either the q function or the

N function is relevant to investment decisions.

For costly reversible investment without fixed costs, whenever the firm observes y

above y+ or below y' , the firm will invest or disinvest accordingly. When the firm

invests or disinvests, y after investment or disinvestment equals the corresponding critical

value, y“ or y" . When y is between y” and y‘ , zero investment, or inaction, is

optimal, and capital stock remains at the same level. Since this model assumes that

installation of investment is instantaneous, we do not observe y above y or below y‘ .
For the naive case, the optimal investment rule (the bold dashed line) is entirely located
left of the costly reversible case (the bold solid line), and two critical values are lower
than those for the costly reversible case.

Figure 1.2 (b) shows optimal investment, dl,, and optimal disinvestment, dD,, as a
function of the ratio of the stochastic variable, Z, to capital stock before investment, K _.
In figure 1.2 (b), a solid line is optimal investment and disinvestment for the costly

reversible investment, and a dashed line is optimal investment for the naive case. Both
lines are derived from figure 1.2 (a). When the ratio, Z, / K,_. is between y+ and y’ , zero

investment is optimal so that K,, = K,_, or d], = O and d0, = 0. If the ratio exceeds y+ ,

optimal investment is positive and given by

.11, =—ZL—K,_. (17)

If the ratio is below y" , optimal disinvestment is positive and given by

d0, = K,_ — 2:. (18)

y

 

 

 

 

 

Investmfm 1"
that. when ‘h‘
“hell ll doc‘S 2
located left of
the firm come
stock. the tint
it does not.

F1 gure.
respectiyely. '
11 heneyerj' mt

inyestment is 0
1p; 1 from belt
intestrn'ent. y"
Opposite moye.

the resale price I

Investment for the naive case is entirely located left of costly reversible investment, so
that, when the firm contemplates future investment, its optimal investment is smaller than
when it does not consider future investment. Disinvestment for the naive case is also
located left of costly reversible investment, so that optimal disinvestment is larger when
the firm contemplates future disinvestment than when it does not. In terms of capital
stock, the firm chooses lower capital stock when it consider future investment than when
it does not.

Figures 1.3 and 1.4 show the optimal rule for investment and disinvestment,
respectively. The optimal rule is to bring y back to one of the two critical values
whenever y moves beyond the corresponding critical value. Otherwise, inaction or zero

investment is optimal. In figure 1.3, the function q approaches the purchase price line

( p}; ) from below, becomes tangent to the purchase price line at the critical point for
investment, y+ , and then moves downward. In figure 1.4, the function q shows the
opposite move. It approaches the resale price line ( p;- ) from above, becomes tangent to

the resale price line at the critical point for disinvestment, y‘ , and then moves upward.

 

 

 

 

 

 

Figure 1.3 Optimal Rule for Investment in
Costly Reversible Investment without Fixed Costs Model

(Hz 0.143, A,z 0.506, a: 0.06, 7: 0.07, #2 = 0.05. 02 = #2. y e 2.01)

 

 

 

 

CJ(Y)
/
/
/
/
0.81L /
‘ /
/
/
/ <1(3’)
0.6 /
/
/
/ -
N(Y)/ y p”
0.4 /
z -
0 0.3 06 Y

Figure 1.4 Optimal Rule for Disinvestment in
Costly Reversible Investment without Fixed Costs Model

(61:: 0.143, A,z 0.506, a: 0.06, y: 0.07,;1z = 0.05. a, = #2. y' e 0.369)

 

 

COS

 

This chapter der
fixed costs. both
y. The inyestme
on one hand. and

marginal analysis

 

reyersible inyestn
We can 111

from intestment v

subj

Here’ F/ZIei and I

Coefficients. F1 an

folloyying Bellman 1

 

 

"

 

 

CHAPTER 2
COSTLY REVERSIBLE INVESTMENT WITHFIXED COSTS

This chapter derives the optimal investment rule when investment has fixed costs. Due to
fixed costs, both investment and disinvestment have a trigger value and a target value in
y. The investment costs have two parts: payment for new capital or revenue from resale
on one hand, and fixed costs on the other hand. Fixed costs, however, do not affect the
marginal analysis of investment, so that the Bellman equation is the same as costly
reversible investment without fixed costs.

We can write the value function, V(K,,Z,), as follows. There is one difference

from investment without fixed costs. i.e., a specification of fixed costs.

V(K,,Z,)= max E,[Ee‘”{A,,Z,'jf’K,”,,ds—1C,,,.}:l (19)

:(IIH. .dl),,,}

subject to dK,,_,. = all,” — dD,,,. — 5 K,,,dt, K,,, 2 0 Vs 2 0,
dZHs = #2 Zl+.i‘dt + 02 ZI+.\'dZZ 9 and
[CH-.1- : [71261],” " pKdDHs + FIZt+x + FDZHx

Here, F, Z,” and F ,,Z,,_,. are fixed costs for investment and disinvestment, respectively.

Coefficients, F, and F D are constant and given. Then, equation (19) yields the

following Bellman equation.
1-0 0 1 2 2
yV(K,,Z,)= AnZ, K, -5 Ky, + pzzyz + 56,2, V2, (20)

The solution of equation (20) is

Vx(- )= q(y) = Hy” + Biyi‘ + Biy‘” - (21)

Here, H, (pi: and (p,. are defined similar to investment without fixed costs.

 

 

 

 

 

 

 

2.]. Optimal
omamsflwtr

.- ,9 .and .‘
.lrr' . "'

 

and
At the trigger ya}

int'esting.

Equal
until function q l"
lllbl. The benel
fixed costs of 1m
disinyestment y 1e

Although

hincnon q \yithou

p. satisfy ti.

q “Ithout fixed c0

hononmga,

lowers. Similarly

,
.1

St
came appendix 1

2-1. Optimal Investment for Costly Reversible Investment with Fixed Costs
Denoting the trigger values and the target values of investment and disinvestment by y}, ,

y}, , y}; , and y,’.u , respectively, the boundary conditions for equation (21) become

qui. )= pi . ’ (22a)

qui..l= pi, (22b)

qui. )= p; . (22c)

qu-I..l= pi. (22d)

Bil” (' l- Pi I“ = ”I , (22c)

and fill); - VK (- )]dK = ZF,, . (221)

At the trigger value for investment, the firm is indifferent between investing and not
investing. Equation (22a) represents this. When the firm invests, the firm should invest
until function q becomes equal to the purchase price of capital. This yields equation
(22b). The benefit from investment from the trigger value to the target value is equal to
fixed costs of investment. Equation (22c) shows this. Similar arguments for
disinvestment yield equations (22c), (22d), and (221).

Although the function q with fixed costs has the same equation form as the
function q without fixed costs, two coefficients for the function q with fixed costs, BL

and BI. , satisfy the following relation with the corresponding coefficients for the function
q without fixed costs, BN and 3,).3

By < B; < 0 < Bi, < BN (23)
By multiplying a positive coefficient smaller than BN, the minimum of the function q

lowers. Similarly, by multiplying a negative coefficient larger than B,., the maximum of

 

3 See the appendix C for proof.

 

 

 

 

the function 4
Values for 1hs‘

conditit‘nS (*2:

From '
r. l-.

and
r... til = -

flc e

r
‘ 315 a \‘CClt

::’Ll'. ‘
v r I \
‘,.'lv"ru. I"

C l"

’ .

equations (24

Figure 2.1
Fe '
he solid line is 11

Til .
thout fixed cost

the function q rises. By choosing appropriate B}, and B}. , the model yields positive
values for the four critical values, y}, . YT, , y-f-u . and y], , satisfying the boundary
conditions (22a) to (220.

From equations (21) and (22a) to (22f), we have

 

 

F. (e) = H01. 1‘" + at (yr-.1" + B: (511,)“ — p: = 0. (24a)
)= )H(yi. )' "+B;(y1..) + 8; (yr-.1” — pi = 0. (24b)
(5):éc:=1‘1(y,,l9+8i,.,‘(y.,)w’wL BM)“ —p;,=0, (24c)
=H(y1‘.1 )1 0 +8 811011,)“ 813(3)], I” - p2 = 0, (24d)
A, 1p. —1 + 1p. -1
F5(5)=::1__—6)[(y1r)0—(yn1)-0+](p8 -(y1-..l l

B' (24c)

+ (011 '_1 [(3% l”—I -(y1§.)""’_'l+(p/01EN[boy/id)"I —(y1*~.)“' l— F1 = 0.

and
131(6) = -p12l(y11.ll -(y13)_' l+ 7&7) [(321 I” -(y1‘-,)”l

(24f)

 

 

— (of: 1lfy-1'~..)“"" —(y11)"’"" - ,1? ”_ 1l(yi.)"”"' — (yr. )‘”"" l— E. = 0.

Here, 6} is a vector of six unknowns for investment with fixed costs, where

6 = {y-,’-,, y},,, y}, , y;,,, B; , B}. }. This analysis again numerically solves the system of
equations (24a) to (241).

Figure 2.1 shows one solution of costly reversible investment with fixed costs.
The solid line is the function q for investment with fixed costs, while a broken line is q
without fixed costs. And, a dashed line is the function N. The difference between q with
fixed costs and without fixed costs is only the inclusion of fixed costs, while all other
conditions are the same. When there are fixed costs, the peak of the function q moves

upward and to the right. As a result, the function q reaches above the purchase price line

20

 

 

 

 

 

1p; :1), An.
mmgponds ltl
same time. the
minimum of th.
betoeen the fur.
or the left hand
ofthe function (1
disinyest. while
in either part (0 1

diff <"‘1‘:J < If” <

 

 

 

FiQUre 2

( p; = l ). And, the crescent area between the function q and the purchase price line
corresponds to fixed costs of investment, or the left hand side of equation (22c). At the

same time, the minimum of the function q moves downward and to the left, and the

minimum of the function q is below the resale price line ( p; =9 0.5 ). A triangular area
between the function q and the resale price corresponds to fixed costs of disinvestment,
or the lefi hand side of equation (22f). The peak of the function q is flat, while the trough
of the function q is steep. When y reaches part (D) in figure 2.1, the firm should
disinvest, while when y reaches part (I) in the figure, the firm should invest. If y remains

in either part (O), (O'), or (0"), then zero investment is optimal. Also, figure 2.1 shows

~ — + +
yo <y-1-., <y-111 <y/‘r-

 

 

 

 

 

Figure 2.1 q(y) and N(y) for Costly Reversible Investment with Fixed Costs

(Hz 0.143, Aflz 0.506, 5: 0.06, 7: 0.07, ,uz = 0.05, 0'; = 0.05,
17;: 19 PE=0-5,F1)=0.5,F,=0.05)

21

 

 

 

 

 

F igurer
respectiyely. l
exceeds the co:
abate 13.}. the
figure 3.2. the 1'
l33-67). Simil.

1110185 below 1.

ft,

 

Figures 2.2 and 2.3 show the optimal rule for investment and disinvestment,
respectively. The optimal rule is to bring y to one of two target values, whenever y

exceeds the corresponding trigger value. For example, the pre-investment level of y is

above y}, , the firm should invest so that the post-investment level of y equals y;,. In
figure 2.2, the optimal investment is to bring y to y}; (z 1.26), whenever y exceeds y},
(z 3.67). Similarly, in figure 2.3, the firm makes y equal to y,‘,, (z 0.352), whenever y

moves below y}, (2 0.150). When y is between y}, and y}, , inaction is optimal.

 

 

 

Figure 2.2 Optimal Rule for Investment in
Costly Reversible Investment with Fixed Costs Model

(6:: 0.143, A,z 0.506, 5: 0.06, y: 0.07, ,le = 0.05, 02 = 0.05,
p7} = 1, p1} = 0.5, F,, = 0.5, F, = 0.05, yiuz 1.26, y}, z 3.67)

22

 

 

 

 

2-2. I'ser Cost

Abel and Ebe

rl y
the User cost of c
Jory ‘er

with fixed costs. I

otter at the trim

lorgenson t l 963)

ere. pis the derit
COIN

~ nt oyer time. .

 

 

 

 

 

Figure 2.3 Optimal Rule for Disinvestment in
Costly Reversible Investment with Fixed Costs Model

(as 0143,14,. z 0.506, 5: 0.06, 7: 007,112: 0.05, 6, = 0.05,
p1: = 1. p12 —- 0.5. F,. = 0.5, F, = 0.05. y], 4 0.150, ym 0.352)

2-2. User Cost of Capital

Abel and Eberly (1996) show that, for costly reversible investment without fixed costs,
the user cost of capital is higher for investment or lower for disinvestment than
Jorgenson’s user cost of capital. This section shows that, for costly reversible investment
with fixed costs, the user cost of capital is even higher at the trigger for investment and
lower at the trigger for disinvestment than those in the Abel and Eberly’s model.

Jorgenson (1963) show that the user cost of capital, c_,, is
c} = (y + 6)p,‘} + [7,} for investment, or (25a)
c] = (7 + §)p,} + p; for disinvestment. (25b)
Here, p is the derivative of p with respect to time. In this analysis, the price of capital is

constant over time, so that

23

 

 

 

 

no and Eb”

Then. the user c

Abel and Eberl y

Because the fixe
yahd for inyesttr.

dtsinyestment th

c} = (y + 5)p,2 for investment, or (26a)

c] = (7 + 5 IP12 for disinvestment. (26b)
Abel and Eberly show that, for costly reversible investment without fixed costs, the user

cost of capital, CO»), is the following.

an o + 6)q(y)- g E14101]
= (7 + 5)q(yl- flyyqul- 10.1.y2q"(y)

2
= A,,t9y"6 (27)

Then, the user cost of capital at each trigger value is

c+ = A,,6(y+ )ng for investment without fixed costs. or (28a)
6' = A,t9(y‘ )H) for disinvestment without fixed costs. (28b)

Abel and Eberly also show
c‘ < cj < c} < c‘. (29)

Because the fixed costs of investment do not affect the marginal analysis, equation (27) is
valid for investment with fixed costs. Thus, at the trigger value of investment or

disinvestment, the user cost of capital is
+ + 1—0 - -
0,, = A,,6(y,,) for investment w1th fixed costs, or (30a)
0,7, = A,,6i(y,'-, )H) for disinvestment with fixed costs. (30b)
Therefore, because y}, < y‘ < y’ < y}, as appendix C shows,
c},<c'(<c] <c,}‘)<c+ <0}. (31)
The user cost of capital at the trigger value for investment is higher with fixed costs than

without fixed costs, while the user cost of capital at the trigger value for disinvestment is

lower with fixed costs than without fixed costs.

24

 

 

 

OPTIM

3.1. Optimal

This section di:
aschematic dia
figure 3.1 is opt
\yith fixed costs
the economic in
optimal inyestm
‘he same leyel a.

inaction locus. s

Figure 3.] Sci

CHAPTER 3

OPTIMAL INVESTMENT RULE AND EFFECTS OF PARAMETERS

3-1. Optimal Investment Rule

This section discusses the optimal investment in terms of capital stock. Figure 3.1 shows
a schematic diagram for costly reversible investment with fixed costs. A solid line in
figure 3.1 is optimal post-investment capital stock for the costly reversible investment

with fixed costs. At time t, given firm’s pre-investment capital stock as K _, if the ratio of

the economic indicator to pre-investment capital stock, Z, / K _, is between y}, and y}, ,

optimal investment is zero gross investment or inaction, so that capital stock remains at
the same level and K,, / K,_ = l. The horizontal locus in figure 3.1, which is called

inaction locus, shows this.

Ktt/Kt_

 

 

 

Figure 3.1 Schematic Diagram for Costly Reversible Investment with Fixed Costs

25

 

 

 

 

 

The up
range by mere.

that

The slope of K
higher value of
loner and the c
inaction locus 2

follows:

Similarl

The upper end of inaction locus is y}; , so that higher y-T-r Widens the inaction

range by increasing its upper limit. When the ratio exceeds y}, , the firm will invest such

that

 

 

 

K,, 1 z, .
= + . (32)
Kt— y’l'u KI-

The slope of K,+ / K,_ locus for investment is proportional to the reciprocal of yin. A

higher value of y}, flattens investment locus, so that post-investment capital stock, K,,, is

lower and the optimal investment is smaller. Due to fixed costs, there is a gap between

inaction locus and investment locus at y;,. Optimal investment can be written as

follows:

d1, = 2’ —K,_ for 2,2 z+(=y;,K,_). (33)

+
ylit

 

Similarly, when the ratio Z, / K,_ moves below y}, the firm disinvests such that

K,+ 1 Z,
-Hs , (3..
K1- y'ra K1—

 

The slope of K,+ / K,- locus for disinvestment is proportional to the reciprocal of y;,,.
There is a similar gap for disinvestment. The lower end of inaction locus is y], , so that
higher yi, narrows the inaction range by decreasing its lower limit. A higher value of

y], flattens disinvestment locus, so that post-investment capital stock is lower and

optimal disinvestment is larger, in contrast to Optimal investment. Optimal disinvestment

can be written as follows:

 

d0, = K,_ - Z_ for Z, 2 Z'( = y;.K,_). (35)
9’71:

When there are no fixed costs of investment, there is no gap among disinvestment

locus, inaction locus and investment locus. In other words. for investment without fixed

26

 

 

 

 

costs. the dis”

G is large 11 he.
costs. minimur
respectiyely. in
capital stock. (
for disinyestme
large GT Cone:-
3-2. Effects of
This section pm
economic paran
lltlmber 0f Simu
tban 3.0.000 11,,
mittbuSt. TTIC
€Xt‘ep1 the {firm

(“cc

parameters such
uncar in the CCOT

Table 3.1

all of the econom
capital. The pure
the differential eq

tenable. #2. the d

l??-
.. O). In other u
ah ‘
0111a redundan
c
3131111

at10ns first f0

costs, the disinvestment locus, inaction locus and investment locus are connected, and
yi‘, = yi. = y+ and y}, = y}. = y'-

The size of the inaction range is measured by G(= Z i / Z ' = y’ /y‘ or y; /y;, ).
G is large when the range of inaction is wide in terms of Z, . For investment with fixed

costs, minimum investment and minimum disinvestment are measured by G+ and G’ ,

respectively, in terms of the ratio of post-investment capital stock to pre-investment
capital stock. G + and G ’ are defined as K,, / K,_ ( y}, /y,+-,, for investment, or y}. /y,'-,,
for disinvestment). A large G+ corresponds to a large value of minimum investment. A

large G’ corresponds to a smaller value of minimum disinvestment.

3-2. Effects of Parameters on Optimal Investment

This section presents simulations constructed under a wide range of values for the
economic parameters in order to investigate their effects on capital investment. The
number of simulation cases is more than 90,000 for investment with fixed costs and more
than 30,000 for investment without fixed costs. Thus, the conclusions of the simulations
are robust. The simulations show that all of the critical values of the optimal investment,
except the trigger value for investment, are approximately linear in the economic
parameters such as the discount rate, while the trigger value for investment is semi-log
linear in the economic parameters.

Table 3.] lists the economic parameters for the simulations. The simulations vary
all of the economic parameters except the depreciation rate and the purchase price of
capital. The purchase price of capital is used as the numeraire. In equation (9), which is
the differential equation for the theoretical model, trend in the economic indicator
variable, #2, the discount rate, 7, and the depreciation rate, 5, appear as #2 + 5 ( E p, ) and
(7+ 6). In other words, three parameters, ,uz, rand 6, yield two coefficients. In order to
avoid a redundancy in the simulations, the simulations drop the depreciation rate. The

simulations first focus on two parameters: lag, and the spread, which is defined as the

27

 

 

,___

l C(I‘él‘l‘lClcn‘i
i—-—'——.—

i Purchase Pl
K
( Trend in Z l
’3

Hand .4- are '
Douglas pdeU
lalues for dirt“
1. 2. and 3. ant
simulation part
the same simul.
rather than tyyet

difference betyyt
lihen the simul:
0fthe critical yal
linear in the mo

Home 3.’

ios-ts and with fix

uni (h) for inyestr
3r . .
eyery flat. As a

l“
”1 let and to. 11m
“fitment. )”

[a . an

The
refore. the linear

{Time
lsér . a
C

Table 3.1 Cases for Simulations

 

 

 

 

 

 

 

 

 

 

 

 

 

Parameter Values Number of Points
Elasticity of Profit (6)‘ 0.06 ~ 0.71 19
Coefficient of Profit ( A. )‘ 0.17 ~ 0.70 q 19
Depreciation Rate ( 5 ) 0.06 l
Volatility in Z, ( 0'2) 0.01, 0.05 and 0.1 3
Purchase Price of Capital ( pg) 1 ( numeraire I 1
Resale Price of Capital (pf, ) 025» 0-5 and 0-75 3
Fixed Costs of Investment (F ,, F ,,) 0.1, 0.5 and 1 3
Trend in Z, ( ,uz) -0.06 to 0.09 by 0.01 16
Spread ( E 7— pg) 0.001 to 0.091 by 0.01 10

 

 

T 6and A, are determined by an iso-elastic demand curve, Q, = (P/ X , )‘S , and a Cobb-
Douglas production function, X 2K “L” , where X , and X, are stochastic coefficients.
Values for three parameters, a, ,6 and 6‘, are determined as a + ,6 = l, 0.8 and 0.6, ,6 / a =
1, 2, and 3, and £= 1.5, 3, and 6. As the three parameters, a, ,Band 8, yield the two
simulation parameters, Band A.., different combinations of the three parameters result in
the same simulation parameters. Thus, the number of the simulation cases is nineteen
rather than twenty seven.

difference between the discount rate and trend in the economic indicator variable, ,uz.
When the Simulation changes the spread and ,uz, and keeps other parameters constant, all
of the critical values except the trigger value for investment with fixed costs are very
linear in the two parameters.

Figure 3.2 shows one example of the critical values for investment without fixed
costs and with fixed costs. Each graph in figure 3.2 contains 139 points. In figure 3.2 (a)
and (b) for investment without fixed costs, the surfaces of the critical values, y+ and y'
are very flat. As a result, the linear approximation is a good fit. Similarly, in figure 3.2

(c), (e) and (f), three critical values for investment with fixed costs: the target value for

investment, yfi, , and the two critical values for disinvestment, y}, , and y], , are very flat.

Therefore, the linear approximation is a good fit for each critical value. However, the

trigger value for investment with fixed costs, yf, , is convex in both ,uz and the spread

28

 

 

 

 

 

 

 

 

 

 

 

. o I I I u 1 v t 1!
F . I n u l 1 u t I v. i
. 1 i I v v u v u
1 - - i l i- t -
. A- .uCnvC—~2mv>:.- 53% 93.3) ~30..~.-.-.U Auwv

highwwwbomwuw 2.210212%) «laserswvxmrev‘ 323.3%

(a) Critical value for investment, y”

 

 

 

_

(b) Target value for investment, y},

(c) Trigger value for investment, y;

 

Costl Reversible with Fixed Costs

29

 

 

 

Figure 3.2 Critical Values for Investment and Disinvestment

0.5, F, = 0.5, FD = 0.5)

0.05, p;=1, pg=

(19:: 0.143, A,z 0.506, 5=0.06, 0'2

 

.’---.‘.—

3.. J:9..C~a3>:._z:v AOL. 03:3; .35.:me A?»

 

w. 30.0 EU Kiwi. .35 Ga .. so» 99%..

Emu)“ vw \5 c 713.0

((1) Critical value for disinvestment, y‘

;—

ible without Fixed Costs

 

Costl Revers

 

j

ersible with Fixed Costs

 

Costly Rev

(t) Trigger value for disinvestment, y;,

(e) Target value for disinvestment, yr},

30

 

 

 

Figure 3 .2 (cont’d)

 

35 figure 3:
the linear aI

approximati

flgures in par
to unity. so th'

By test
equal to zero. 1

hl‘l‘OIhesis u. it}

ll .
ere, flout: ar.

11 .
ellOmlnalOr is

  
  
 

 

‘allables. Tap:
[elem The h\p( ,

3' .

as figure 3.2 ((1) shows. As a result, the semi-log linear approximation is a better fit than

the linear approximation. With the parameters for figure 3.2, the linear or semi-log linear

approximations of all six critical values and their R2 are as follows:

y+ =0.775+18.4><spread+l7.6x,uz (R2 =0.999, data size = 139)
(0.00407) (0.0655) (0.0428)

y' =0.109+6.12xspread+2.79xyz (R2 =0.997, data size = 139)
(0.00580) (0.0934) (0.0610)

yi. =0.350+12.0xspread+4.61 x112 (R2 =0.987, data size = 139)
(0.00937) (0.151) (0.0985)

Ln yr. =1 .09+l7.9xspread+l4.9x,uz (R2 =0.982, data size =139)
(0.0198) (0.318) (0.208)

y},=0.0995+5.92xspread+1.65xyz (R2 =0.990, data size =139)
(0.0107)(O.171) (0.112)

Yr} =0.0774le .67xspread+0.488x,uz (R2 =0.990, data size =139)
(0.00742) (0.1 19) (0.0780)

Figures in parentheses are the standard errors. The R2 of all critical values are very close

to unity, so that the linear approximations are a good fit with regard to ,uzand the spread.
By testing the hypothesis that the sum of the coefficients of the spread and ,uz are

equal to zero, the analysis investigates the significance of 7. When yis significant, the

hypothesis will be rejected. The test statistics is

A

+
flSPTead flit/WT ~ I data size—k - (3 6)

se(fl5prcad + 1610 )

[:

 

Here, Bspmad and ,6”, are estimated coefficients for the spread and [13, respectively. The

denominator is the standard error of ( AM, + .3111 ), and k is the number of explanatory

variables. Table 3.2 shows the t-statistics under the hypothesis. All of six tested values
reject the hypothesis of insignificant rat the one percent level. Therefore, the

simulations conclude that the discount rate, y, is statistically significant.

31

 

Simila
Tables 3.3 to I
models with 111
without interat
approximation
spread. 0,. and
for the quadrant
constant. the set
and at most 28 f
a? higher than (
a”31.1515 includes
Ills Case of the on

R: higher than 0.

 

Table 3.2 Significance of Discount Rate

 

t-Statistics

Critical Value of Investment, y+ 4179*”
Critical Value of Disinvestment, y‘ 726*”
Target Value of Investment, y;,, 64.] *iluk
Trigger Value of Investment, Ln y}; 2355*”
Target Value of Disinvestment, y}, 3 3.6“”:
Target Value of Disinvestment, y}, 133*”

 

***: reject H0 at 1%

Similarly, the critical values show a linear relationship to the other parameters.
Tables 3.3 to 3.8 show estimated coefficients of five approximations: two for linear
models with and without interaction terms, and three for quadratic models with and

without interaction terms. There are eight explanatory variables for the linear
approximation of investment with fixed costs: a constant, 6, A,, p;- , fixed costs, the

spread, 0'2, and ,uz, and seven for investment without fixed costs. Explanatory variables
for the quadratic approximation are at most 36 for investment with fixed costs: the
constant, the seven variables, squares of the seven variables, and 21 interaction terms,
and at most 28 for investment without fixed costs. Even the linear approximations have
R2 higher than 0.7, although they have only seven or eight explanatory variables. As the
analysis includes more explanatory variables such as squared terms, the R2 increases. In
the case of the quadratic approximation with all interaction terms, all critical values have

R2 higher than 0.9.

32

 

 

 

Table 3

constant .3

 

1

Q .9 LL. h- Q3 %
Rd 11.; I;

I

¢J

Pi;

r
ffx. A,
fix. 0:.
9* Pi
.~l,x 0}
.i.x m.
Bxpi
Spread
Spreadj _,
9" Spread X
4" Spread N ,
“Spread N
p; x SDread N
4" l7.

:zzzzzzrer

Z7

11.3

7.

44:x 02

0:111.
‘c’.

 

Table 3.3 Critical Value for Investment without Fixed Costs, y+

Linear with Quadratic with Quadratic with

 

 

Linear Quadratic Interaction Interaction( 1) Interaction ( 2)
constant 3.7 (0.03) 7.8 (0.04) 1.4 (0.06) 5.5 (0.05) -0.7 (0.11)
19 -5.6 (0.03) -15.7 (0.07) -3.0 (0.07) -13.01(0.07) 1.4 (0.24)
192 N/A 12.5 (0.09) N/A 12.5 (0.08) 5.0 (0.14)
A, -3.8 (0.04) -15.0 (0.12) 0.0 (0.08) -11.3 (0.12) 8.9 (0.34)
A; N/A 10.7 (0.13) N/A 10.7 (0.12) -5.1 (0.27)
oz 1.3 (0.10) 0.2 (0.31) 0.8 (0.21) -0.2 (0.30) 1.9 (0.48)
6; N/A 9.1 (2.67) N/A 9.0 (2.36) 9.0 (2.21)
p; -0.0 (0.02) -0.0 (0.09) 0.0 (0.04) 0.0 (0.08) -00 (0.10)
pg? N/A 0.0 (0.09) N/A 0.0 (0.08) 0.0 (0.07)
19x11, N/A N/A N/A N/A -22.9 (0.36)
6x 6, N/A N/A N/A N/A -40 (0.48)
9x p; N/A N/A N/A N/A 0.0 (0.09)
A,x 02 N/A N/A N/A N/A -2.3 (0.60)
14,,pr N/A N/A N/A N/A 0.0 (0.11)
0'2pr N/A N/A N/A N/A 0.0 (0.28)
spread 17.6 (0.13) 17.9 (0.35) 49.3 (0.90) 49.5 (0.66) 46.2 (0.63)
spread2 N/A -1.8 (3.56) N/A -1.9 (3.14) 15.9 (3.05)
axspread N/A N/A -345 (1.06) -359 (0.69) -36.2 (0.65)
A,x spread N/A N/A -52.6 (1.33) -51.1 (0.87) -50.8 (0.81)
azxspread N/A N/A 11.7 (3.43) 10.8 (2.24) 11.2 (2.10)
pgxspreadN/A N/A - -0.3 (0.62) -0.4 (0.40) -0.3 (0.38)
,uz 17.3 (0.09) 16.6 (0.09) 47.6 (0.59) 46.9 (0.39) 43.8 (0.39)
113 N/A 17.9 (1.47) N/A 17.9 (1.30) 28.6 (1.32)
pr2 N/A N/A -359 (0.70) -36.4 (0.46) -36.4 (0.43)
A,xor, N/A N/A -46.2 (0.87) -46.1 (0.57) -459 (0.53)
0'2po N/A N/A -5.7 (2.26) -4.4 (1.47) -4.5 (1.38)
1),;an N/A N/A -0.2 (0.41) 01 (0.27) -0.1 (0.25)
spread x 11, N/A N/A N/A N/A 44.9 (2.17)
R2 0.717 0.862 0.748 0.893 0.906 '

 

data size: 33,143
standard error in parenthesis

33

 

 

Tal

constant

a -
62 .‘\
.i, -l
if N

0"; X
h; 1

Pi: .\
9&4, }\

\.
fan; 1\'.

v
.
v

-

8x 0:,

a.

.i,x 0:,

0': X p,L ,1
Spread 5.1.

.\
.1.ng \ :1
.\

Spread: N A
[4* Spread N A
rig. Spread NA
QfXSPread N A
p; *- spread N A
”4’ 2.95
”I: N A
61 #3 N, A
1,1,3 \' A
57:1 Nu. NA
19,7 x #2 NA
Fiend x #2 N A

i3143i
ZCI 33 14
Stan . ‘ 3
dud enoi in par

 

Table 3.4 Critical Value for Disinvestment without Fixed Costs, y“

Linear with Quadratic with Quadratic with

 

 

Linear Quadratic Interaction Interaction ( 1 ) Interaction ( 2)
constant 0.21 (0.01) 1.22 (0.02) 0.03 (0.02) 1.06 (0.02) -0.88 (0.04)
6? -l.51 (0.01) -4.28 (0.03) -0.50 (0.02) -3.28 (0.03) 0.78 (0.07)
62 N/A 3.50 (0.04) N/A 3.53 (0.03) 2.43 (0.04)
A, -0.65 (0.01) -2.12 (0.05) 0.13 (0.03) —1.37 (0.05) 2.70 (0.11)
A; N/A 1.14 (0.06) N/A 1.13 (0.05) -1.20 (0.09)
a; -0.11 (0.03) 0.03 (0.13) -0.06 (0.07) 0.03 (0.12) -0.07 (0.15)
0; N/A -1.23 (1.15) N/A -0.88 (0.93) -090 (0.69)
pf, 1.18 (0.01) 0.14 (0.04) 0.30 (0.01) —0.75 (0.03) 1.35 (0.03)
pf N/A 1.04 (0.04) N/A 1.05 (0.03) 1.05 (0.02)
6x A, N/A N/A N/A N/A -3.37 (0.11)
6x 02 N/A N/A N/A N/A 0.44 (0.15)
0x p; N/A N/A N/A N/A -3.98 (0.03)
A, x 02 N/A N/A N/A N/A 0.20 (0.19)
A, x p}, N/A N/A N/A N/A -2.27 (0.03)
0'; x pg. N/A N/A N/A N/A -0.21 (0.09)
spread 5.03 (0.05) 4.65 (0.15) 8.32 (0.28) 7.87 (0.26) 7.31 (0.20)
spread2 N/A 4.69 (1.53) N/A 4.87 (1.23) 7.39 (0.95)
6x spread N/A N/A —l6.10 (0.34) -16.50 (0.27) -I6.40 (0.20)
A, x spread N/A N/A -11.90 (0.42) -11.50 (0.34) -11.40 (0.25)
02 x spread N/A N/A -2.09 (1.09) -2.27 (0.88) -2.38 (0.65)
p; x spreadN/A N/A 13.00 (0.20) 13.00 (0.16) 13.20 (0.12)
,uz 2.95 (0.03) 2.92 (0.04) 3.35 (0.19) 3.28 (0.15) 2.97 (0.12)
p; N/A 0.74 (0.63) N/A 1.66 (0.51) 2.98 (0.41)
6x #2 N/A N/A -8.30 (0.22) -8.42 (0.18) -8.56 (0.13)
A, x ,uz N/A N/A -7.16 (0.28) -7.16 (0.22) -7.22 (0.17)
0'2 x #2 N/A N/A 1.87 (0.72) 2.11 (0.58) 2.09 (0.43)
p; x #2 N/A N/A 9.81 (0.13) 9.84 (0.10) 9.88 (0.08)
spread x #2 N/A N/A N/A N/A 5.36 (0.68)
R2 0.709 0.783 0.784 0.859 0.922

 

data size: 33,143
standard error in parenthesis

34

 

__________________
11011513111

9
a: T
.4. -

2 x
.1,
0':
a; :\

‘—

p;

PI
tned costs
it2

91. .1,

6x a,

9* 17;

9x fc

:1 .x a,

:1, x p;

.1, x fc

G: X p;

0: X fc

Fix It
Spread
Sprfadl .‘
9* Spread x ,1
i= " Spread N g
‘7: “ Spread N;
If}. 7“ Spread N ".4

1C ’* Spread \‘3:
.“2

A].

gzzzzzzzzxzz

ix:

A

u2 .
‘1 AA
"1

H! #2

.1 x L

' ‘X #2 N

6; X [“2 \,
\

Table 3.5 Target Value for Investment with Fixed Costs, y;

Linear with Quadratic with Quadratic with

 

 

Linear Quadratic Interaction Interaction( 1 )Interaction ( 2)
constant 2.0 (0.01) 3.7 (0.01) 0.7 (0.01) 2.4 (0.01) 0.6 (0.03)
a -2.8 (0.01) -7.2 (0.02) -1.2 (0.02) -5.5. (0.02) -1.2 (0.06)
(92 N/A 5.5 (0.02) N/A 5.3 (0.02) 2.8 (0.03)
A, -1.7 (0.01) -5.8 (0.03) -0.1 (0.02) -4.2 (0.03) 2.2 (0.08)
A; N/A 3.8 (0.03) N/A 3.8 (0.03) -1.4 (0.06)
oz 0.3 (0.02) 0.1 (0.08) 0.3 (0.05) 0.0 (0.07) 0.7 (0.11)
0; N/A 2.2 (0.68) N/A 2.3 (0.55) 2.3 (0.50)
p; —0.0 (0.00) -0.0 (0.02) -0.0 (0.01) -0.0 (0.02) -0.0 (0.02)
p,2 N/A 0.0 (0.02) N/A 0.0 (0.02) 0.0 (0.02)
fixedcosts -0.3 (0.00) -0.8 (0.01) -0.1 (0.00) -0.6 (0.01) -1.1 (0.01)
fc2 N/A 0.4 (0.01) N/A 0.4 (0.01) 0.4 (0.01)
am, N/A N/A N/A N/A -7.8 (0.08)
9x 0, N/A N/A N/A N/A -0.9 (0.11)
9x p; N/A N/A N/A N/A -0.0 (0.02)
axfc N/A N/A N/A N/A 1.0 (0.01)
A,x 0. N/A N/A N/A N/A -0.4 (0.13)
A,x p; N/A N/A N/A N/A -0.0 (0.02)
A,x fc N/A N/A N/A N/A 0.5 (0.01)
a, x p, N/A N/A N/A N/A -0.0 (0.06)
oz x fc N/A N/A N/A N/A -0.3 (0.04)
pgxfc N/A N/A N/A N/A 0.0 (0.01)
spread 10.4 (0.03) 8.6 (0.09) 30.4 (0.22) 28.3 (0.15) 28.2 (0.14)
spreadz N/A 23.5 (0.91) N/A 22.9 (0.72) 21.2 (0.69)
axspread N/A N/A -301 (0.26) -28.8 (0.16) -27.1 (0.15)
A,xspread N/A N/A -252 (0.31) -26.0 (0.20) -25.7 (0.18)
azxspread N/A N/A 0.7 (0.81) 0.8 (0.52) 0.3 (0.48)
pgxspreadN/A N/A 0.2 (0.15) -0.1 (0.09) -0.1 (0.09)
fcxspread N/A N/A -1.7 (0.08) -1.0 (0.05) -0.7 (0.05)
p, 5.1 (0.02) 5.3 (0.02) 15.8 (0.15) 15.0 (0.09) 15.4 (0.09)
12; N/A 3.2 (0.39) N/A 5.4 (0.32) 3.7 (0.31)
9ng N/A N/A -9.4 (0.17) -8.2 (0.11) -7.8 (0.10)
A,xyz N/A N/A -12.8 (0.21) -124 (0.14) -123 (0.13)
02po N/A N/A -1.4 (0.55) -O.8 (0.35) -0.8 (0.32)
,7;po N/A N/A -0.1 (0.10) 0.0 (0.06) -0.0 (0.06)
fox #2 N/A N/A -4.8 (0.06) -4.2 (0.04) -4.3 (0.03)
spread x 1., N/A N/A N/A N/A -8.0 (0.50)
R2 0.763 0.881 0.812 0.924 0.936

 

data size: 91,101
standard error in parenthesis

35

 

 

 

I
Table 3.6

60115111111 ‘
(9 -
193 N
.1. -1
.1; N
a 1
a; x
P; -0
pi" N

fixed costs 3
f0

1\'
9x4: \'
9110:. \-
8x1); \-
th‘c .\"
.1,x 0'2 N,
.‘i,)( {C \.
0:ng x.
07X 10 N.
pix fc \
Spread :1.
Spread: \~

.1
a
1
I:
l

‘4’! Spread N
Oi" Spread N

i?” x Spread N
.u. .

 

Table 3.6 Trigger Value for Investment with Fixed Costs (Natural Logged), Ln y;

Linear with Quadratic withQuadratic with

 

 

Linear Quadratic Interaction Interaction (1) Interaction (2)
constant 4.1 (0.02) 8.4 (0.03) 4.4 (0.04) 9.1 (0.03) -3.7 (0.07)
6? -7.4 (0.02) -10.8 (0.06) -8.8 (0.04) -12.8_ (0.06) 14.3 (0.15)
62 N/A 3.8 (0.07) N/A 4.3 (0.06) -92 (0.09)
A, -6.2 (0.03) -25.1 (0.10) -4.6 (0.05) -24.3 (0.09) 13.2 (0.21)
A: N/A 20.4 (0.11) N/A 21.5 (0.09) -5.6 (0.17)
0'; 1.2 (0.07) 0.2 (0.26) 1.6 (0.14) 0.4 (0.22) 0.7 (0.31)
0; N/A 9.7 (2.24) N/A 9.7 (1.75) 9.6 (1.39)
p; -0.0 (0.01) 0.0 (0.07) -0.0 (0.02) -0.0 (0.06) 0.0 (0.06)
p,}2 N/A -0.0 (0.07) N/A -0.0 (0.06) 0.0 (0.04)
fixed costs 2.0 (0.01) 2.8 (0.03) 0.7 (0.01) 1.4 (0.02) 4.4 (0.03)
fc2 N/A -0.7 (0.02) N/A -0.6 (0.02) -0.6 (0.01)
19x A, N/A N/A N/A N/A -41.2 (0.23)
6x 02 N/A N/A N/A N/A -0.4 (0.30)
Exp; N/A N/A N/A N/A 0.0 (0.05)
foc N/A N/A N/A N/A -2.9 (0.03)
A,xoz N/A N/A N/A N/A -0.8 (0.37)
A,xp,‘< N/A N/A N/A N/A -0.0 (0.07)
A,xfc N/A N/A N/A N/A -5.1 (0.04)
azxp; N/A N/A N/A N/A -0.0 (0.18)
ozxfc N/A N/A N/A N/A 0.5 (0.10)
pgxfc 7 N/A N/A N/A N/A -0.0 (0.02)
spread 22.4 (0.09) 21.5 (0.29) 17.5 (0.60) 17.4 (0.49) 25.4 (0.40)
spread2 N/A 19.5 (2.98) N/A 31.0 (2.34) -0.8 (1.92)
9x spread N/A N/A 17.1 (0.71) 19.2 (0.52) 18.6 (0.42)
A,x spread N/A N/A -25.8 (0.87) -32.3 (0.64) -35.2 (0.51)
azxspread N/A N/A 1.9 (2.26) 2.9 (1.66) 1.8 (1.32)
pgxspreadN/A N/A 0.5 (0.41) 0.3 (0.30) 0.0 (0.24)
fcxspread N/A N/A 21.5 (0.23) 22.3 (0.17) 22.4 (0.13)
,uz 19.0 (0.06) 20.1 (0.08) 11.3 (0.40) 11.7 (0.30) 17.7 (0.26)
[2; N/A -13.6 (1.29) N/A -18.9 (1.01) -38.4 (0.87)
19po N/A N/A 32.7 (0.47) 35.4 (0.35) 35.2 (0.28)
A,xluz N/A N/A -12.8 (0.59) -13.5 (0.44) -12.8 (0.35)
0’2po N/A N/A -14.7 (1.53) -14.3 (1.13) -13.6 (0.90)
p,‘, xyz N/A N/A -0.1 (0.28) -0.1 (0.20) -0.0 (0.16)
fcxyz N/A N/A 10.3 (0.15) 11.2 (0.11) 10.9 (0.09)
spread xpz N/A N/A N/A N/A -95.6 (1.40)

R2 0.779 0.849 0.829 0.907 0.941

 

data size: 91,101
standard error in parenthesis

36

 

COHSIJIII

.Q L“ in CD CD
J ’1 I) ‘4 I.‘

0.?

17;

pi:

fixed costs
it:

QXAX

(9x a;
prg

9x fc

.1,x 0’,

:1: x [’1' \
Azx 1C ‘\

@Xp,
CHIC

P; X fc

Spread .
Spread: )(
9" Spread x
‘44 X Spread x
0,). Spread N
94‘ X Spread N
to x Spread N
M l

"d rd .._4

’7/

 

Table 3.7 Target Value for Disinvestment with Fixed Costs, y},

Linear with Quadratic with Quadratic with

 

 

Linear Quadratic Interaction Interaction (1) Interaction (2)

constant 0.24 (0.00) 0.96 (0.01) 0.03 (0.01) 0.75 (0.01) -0.60 (0.02)
6 -1.25 (0.00) -3.37 (0.01) -0.35 (0.01) -2.45 (0.01) 0.16 (0.03)
62 N/A 2.70 (0.02) N/A 2.64 (0.01) 1.99 (0.02)
A, -0.53 (0.01) -1.64 (0.02) ' 0.07 (0.01) -0.97 (0.02) 1.60 (0.04)
A; N/A 0.83 (0.03) N/A 0.75 (0.02) -0.61 (0.04)
02 0.08 (0.02) 0.03 (0.06) 0.08 (0.03) 0.02 (0.05) 0.08 (0.06)
0; N/A 0.48 (0.51) N/A 0.54 (0.40) 0.54 (0.29)
p; -0.92 (0.00) 0.43 (0.02) 0.24 (0.01) -0.23 (0.01) 1.44 (0.01)
p,}2 N/A 0.50 (0.02) N/A 0.48 (0.01) 0.49 (0.01)
fixed costs -0.06 (0.00) -0.14 (0.01) -0.01 (0.00) -0.09 (0.01) -0.07 (0.01)
fc2 N/A 0.07 (0.01) N/A 0.06 (0.00) 0.07 (0.00)
6 x A, N/A N/A N/A N/A -2.07 (0.05)
6x 0'2 N/A N/A N/A N/A -0.32 (0.06)
6x p; N/A N/A N/A N/A -2.96 (0.01)
(9x fc N/A N/A N/A N/A 0.24 (0.01)
A, x o-Z N/A N/A N/A N/A -0.13 (0.08)
A, x p; N/A N/A N/A N/A -1.71 (0.01)
A, x fc N/A N/A N/A N/A 0.05 (0.01)
0'2 x p; N/A N/A N/A N/A 0.25 (0.04)
oz x fc N/A N/A N/A N/A -0.08 (0.02)
p; x fc N/A N/A N/A N/A "-0.24 (0.00)
spread 4.54 (0.02) 3.88 (0.07) 7.96 (0.13) 7.18 (0.11) 7.38 (0.08)
spread2 N/A 8.78 (0.68) N/A 7.79 (0.54) 7.37 (0.40)
6x spread N/A N/A -16.70 (0.15) -16.20 (0.12) -15.60 (0.09)
A, x spread N/A N/A -9.72 (0.18) -9.80 (0.15) -9.74 (0.11)
02 x spread N/A N/A -0.13 (0.47) -0.02 (0.38) -0.16 (0.28)
p; x spreadN/A N/A 11.50 (0.08) 11.40 (0.07) 10.90 (0.05)
fc x spread N/A N/A -0.59 (0.05) -0.31 (0.04) -0.23 (0.03)

,uz 1.59 (0.01) 1.63 (0.02) 3.25 (0.08) 2.82 (0.07) 2.99 (0.05)
p; N/A 2.00 (0.29) N/A 2.70 (0.23) 2.25 (0.18)
0x #2 N/A N/A -4.67 (0.10) -4.17 (0.08) -4.23 (0.06)
A, x ,uz N/A N/A -4.79 (0.12) -4.55 (0.10) -4.59 (0.07)
0'2 x [.12 N/A N/A 0.05 (0.32) 0.26 (0.26) 0.39 (0.19)
p; x #2 N/A N/A 4.72 (0.06) 4.79 (0.05) 4.93 (0.03)

fc x #2 N/A N/A -1.29 (0.03) -0.99 (0.03) -1.00 (0.02)

spread xpz N/A N/A N/A N/A -3.32 (0.29)

RT 0.742 0.808 0.817 0.880 0.938

 

data size: 91,101
standard error in parenthesis

37

 

 

Tal

9 -U
r N
‘4: -0
.1; N
0': -U
0“; N
p; 0
m: N-

tixcd costs -0.

1C: N .
9x .4, N .
91 a. N .-
(ix p; _\' ‘.
(9x fc N .-
1.x 0'; N :
:1. x pA N
M fc N‘ ..
904 x.
(7:! ft N '.~‘
P. X 10 N’ ‘:
spread 1“
Spread N1
9“ Spread N' .1
1 ’Spread N .1
0; ”Dread N‘ .1
p ‘ X SpreadN' .1
“wmsd\1
.U:

: 0‘6
'1 #2: \':
.1.x ‘ A
3 X .113 \ 2x
P~ x i
’1 #2 T'
fC X #7 SA
Spread)< ‘ .' A
R‘ M? J\ A

0.7:
SIZ

Table 3.8 Trigger Value for Disinvestment with Fixed Costs, y},

Linear with Quadratic with Quadratic with

 

 

Linear Quadratic Interaction Interaction (1) Interaction (2)

constant 0.17 (0.00) 0.44 (0.00) 0.07 (0.00) 0.33 (0.00) -0.10 (0.01)
(9 -0.50 (0.00) -1.17 (0.01) -0.22 (0.00) -0.87 (0.01) -0.20 (0.01)
6’2 N/A 0.85 (0.01) N/A 0.81 (0.01) 0.60 (0.01)
A ,, -0.20 (0.00) -0.63 (0.01) -0.02 (0.01) -0.42 (0.01) 0.35 (0.02)
A; N/A 0.35 (0.01) N/A 0.32 (0.01) -0.14 (0.01)
oz -0.09 (0.01) -0.01 (0.03) -0.06 (0.01) 0.01 (0.03) -0.05 (0.03)
0': N/A -0.73 (0.24) N/A -0.65 (0.21) -0.66 (0.13)
p; 0.39 (0.00) 0.23 (0.01) 0.16 (0.00) 0.00 (0.01) 0.78 (0.01)
p,}2 N/A 0.16 (0.01) N/A 0.16 (0.01) 0.16 (0.00)
fixed costs -0.14 (0.00) -0.33 (0.00) -0.03 (0.00) -0.22 (0.00) -0.27 (0.00)
fc2 N/A 0.17 (0.00) N/A 0.16 (0.00) 0.17 (0.00)
0 x A, N/A N/A N/A N/A -0.78 (0.02)
6x 02 N/A N/A N/A N/A 0.32 (0.03)
9x p; N/A N/A N/A N/A -1.08 (0.01)
6x fc N/A N/A N/A N/A 0.52 (0.00)
A,x oz N/A N/A N/A N/A 0.11 (0.03)
A,r x p; N/A N/A N/A N/A -0.57 (0.01)
A,r x fc N/A N/A N/A N/A 0.20 (0.00)
oz x p; N/A N/A N/A N/A -0.21 (0.02)
0'2 x fc N/A N/A N/A N/A 0.06 (0.01)
p; x fc N/A N/A N/A N/A 40.39 (0.00)
spread 1.55 (0.01) 1.54 (0.03) 3.36 (0.06) 3.35 (0.06) 3.26 (0.04)
spread2 N/A 0.84 (0.32) N/A -0.51 (0.28) -0.87 (0.18)
61x spread N/A N/A -5.30 (0.07) -5.06 (0.06) -4.16 (0.04)
A,r x spread N/A N/A -2.78 (0.09) -2.78 (0.08) -2.56 (0.05)
0'2 x spread N/A N/A -1.36 (0.22) -l.28 (0.20) -1.29 (0.12)
p; x spreadN/A N/A 3.83 (0.04) 3.80 (0.04) 3.44 (0.02)

fc x spread N/A N/A -1.90 (0.02) -1.77 (0.02) -1.61 (0.01)

,uz 0.67 (0.01) 0.74 (0.01) 1.21 (0.04) 1.11 (0.04) 1.17 (0.02)
,ug N/A -0.57 (0.14) N/A -0.16 (0.12) -0.72 (0.08)
61x #2 N/A N/A -l.51 (0.05) -1.31 (0.04) -1.12 (0.03)
A,r x #2 N/A N/A -1.44 (0.06) -1.33 (0.05) -l.28 (0.03)
oz x #2 N/A N/A 1.32 (0.15) 1.29 (0.14) 1.28 (0.08)
p; x ,uz N/A N/A 1.77 (0.03) 1.79 (0.02) 1.85 (0.02)

fc x ,uz N/A N/A -0.91 (0.02) -0.82 (0.01) -0.85 (0.01)

spread x ,uz N/A N/A N/A N/A -2.40 (0.13)

R2 0.730 0.778 0.791 0.834 0.936

 

data size: 91,101
standard error in parenthesis

38

 

 

Table
for the appro
interaction 1"
approximall“

the firms are

Here. m 15 [he
explanatory V"
data size 15 ll”
linear appm‘“
linear appFOXll
the interaction
disinvestment-
adclcd to the qU
set of interactio
set. As the quai
all critical value
In additn
log linear. Tablt

empirical rcsean

 

Table 3.9 shows the significance of the quadratic terms and the interaction terms

for the approximation. This analysis tests the significance of the quadratic terms or the

interaction terms by a test comparing the R2. Under the null hypothesis that the
approximation with the quadratic or the interaction terms and the approximation without

the terms are equivalent, the test statistics is

2 2

F : (RWith terms — Rwithouttenns )/m
(1 — Rim. terms )/(data size — k)

 

~ F (m, data size — k). (37) :5.

Here, m is the number of the quadratic or interaction terms, and k is the number of

 

explanatory variables in the model with the quadratic or interaction terms. Because the

data size is large, differences in the R2 for all cases are statistically significant. For the E1
linear approximation, either the quadratic terms or the interaction terms are added to the i
linear approximation. Table 3.9 shows that the quadratic terms are more significant than

the interaction terms for investment, but both terms are equally significant for

disinvestment. For the quadratic approximation, two sets of the interaction terms are

added to the quadratic approximation and the second set of interactions includes the first

set of interactions and additional terms, i.e., the first set is a proper subset of the second
set. As the quadratic approximation includes more interaction terms, its R2 increases for
all critical values. And, the increase of the R2 is statistically significant.

In addition, the three ratios of the critical values, G+ , G" and G, are also semi-
log linear. Tables 3.10 to 3.12 show their approximations. This result facilitates

empirical research.

39

 

l. C0511.V R‘
‘3, F.Statist
Approximali
‘ Model
Linear

 

 

1b) F-Statisti
Approximatit
Model
Linear
Quadratic
\

 

3. Costly Rev
(c) F-Statistic.
Approximatiot
t Model
Linear
Quadratic
\
1d) F-Statistics
[Approximation
‘ Model
1 Linear .‘
Quadratic i

let F—Statistics 1
Approximation '
' Model 1
Linear .
Quadratic i
\
(11 F-Statisties ft
I'Pproximation. ,

ml

1 ”7‘39 H0 at

l

 

Table 3.9 Significance of Quadratic Terms and Interaction Terms

1. Costly Reversible Investment without Fixed Costs

(a) F -Statistics for Critical Value of Investment, y+

 

 

 

 

 

 

 

 

Approximation Quadratic Terms Interactions for Interactions for Interactions for
Model Linear Model Quad. Model (1) Quad. Model (2)
Linear 5,802*** 509*** N/A N/A

Quadratic N/A N/A 1,200*** 1,033***

 

(b) F -Statistics for Critical Value of Disinvestment, y'

 

 

 

 

 

 

 

 

Approximation Quadratic Terms Interactions for Interactions for Interactions for
Model Linear Model Quad. Model (1 ) Quad. Model (2)
Linear l,438*** 1,883*** N/A N/A

Quadratic N/A N/A 2,232*** 3,934***

 

2. Costly Reversible Investment with Fixed Costs

(c) F -Statistics for Target Value of Investment, y};

 

 

 

 

 

 

 

 

Approximation Quadratic Terms Interactions for Interactions for Interactions for
Model Linear Model Quad. Model (1 ) Quad. Model (2)
Linear 12,903*** 2,374*** N/A N/A

Quadratic N/A N/A 5,153*** 3,727***

 

(d) F -Statistics for Trigger Value of Investment, Ln y-f,

 

 

 

 

 

 

 

 

Approximation Quadratic Terms Interactions for Interactions for Interactions for
Model Linear Model Quad. Model (1 ) Quad. Model (2)
Linear 6,032*** 2,663*** N/A N/A

Quadratic N/ A N/A 5,680* * * 6,762* * *

 

(e) F -Statistics for Target Value of Disinvestment, yfi,

 

 

 

 

 

 

 

 

Approximation Quadratic Terms Interactions for Interactions for Interactions for
Model Linear Model Quad. Model (1) Quad. Model (2)
Linear 4,473*** 3,733*** N/A N/A

Quadratic N/A N/A 5,465*** 9,093***

 

(t) F -Statistics for Critical Value of Investment, y;

 

 

 

 

 

 

 

 

Approximation Quadratic Terms Interactions for Interactions for Interactions for
Model Linear Model Quad. Model (1 ) Quad. Model (2)
Linear 2,813*** 2,658*** N/A N/A

Quadratic N/A N/A 3,072*** 10,706***

 

***: reject H0 at 1%

40

 

 

 

 

 

 

 

 

 

Tablt

constant

6’

(9: 1‘
.1, -
.1; .\
0’3 1
013 N
Pi -<
pi‘ .\'
fixed costs I
fc3 N
M. N
91 a: N
0x fc N
.Il,x 0:? N
.11,ng N .
14.x fc N .
a, 1p; N t.
02 X fc N .4,
Pi X fc N

Spread 9“
Spread: N
6x Spread N":
l " SDread N
0‘3 X Spread \‘_,1

 

Table 3.10 Measure of Minimum Investment (Natural Logged), LnG+

Linear with Quadratic withQuadratic with

 

 

Linear Quadratic Interaction Interaction (1) Interaction (2)
constant 2.4 (0.01) 4.9 (0.03) 2.2 (0.03) 4.9 (0.03) -2.0 (0.07)
19 -2.5 (0.02) -5.1 (0.05) -2.6 (0.03) -5.6 (0.05) 8.3 (0.14)
62 N/A 3.0 (0.06) N/A 3.5 (0.05) -3.0 (0.08)
A,r -3.3 (0.02) -13.9 (0.08) -1.4 (0.04) -12.4 (0.07) 6.9 (0.20)
A: N/A 11.3 (0.09) N/A 12.0 (0.08) -0.8 (0.16)
oz 0.8 (0.05) 0.1 (0.21) 0.9 (0.10) 0.1 (0.19) 0.4 (0.28)
0; N/A 6.4 (1.80) N/A 6.5 (1.47) 6.3 (1.27)
pf, -0.0 (0.01) 0.0 (0.06) -0.0 (0.02) -0.0 (0.05) 0.0 (0.06)
pg? N/A -0.0 (0.06) N/A 0.0 (0.05) 0.0 (0.04)
fixedcosts 2.3 (0.01) 3.5 (0.02) 1.1 (0.01) 2.2 (0.02) 5.2 (0.03)
fc2 N/A -1.0 (0.01) N/A -1.0 (0.01) -1.0 (0.01)
6xA, N/A N/A N/A N/A -19.8 (0.21)
6x 02 N/A N/A N/A N/A -0.7 (0.28)
prg N/A N/A N/A N/A -0.0 (0.05)
6xfc N/A N/A N/A N/A -3.0 (0.03)
A,xo'z N/A N/A N/A N/A -1.0 (0.34)
A,xp; N/A N/A N/A N/A -0.0 (0.06)
A,xfc N/A N/A N/A N/A -5.1 (0.03)
azxp; N/A N/A N/A N/A 0.0 (0.16)
ozxfc N/A N/A N/A N/A 0.7 (0.09)
pgxfc N/A N/A N/A N/A 40.0 (0.02)
spread 9.3 (0.07) 6.8 (0.23) 11.4 (0.49) 8.9 (0.41) 11.2 (0.37)
spread2 N/A 32.5 (2.39) N/A 45.8 (1.96) 45.4 (1.76)
19x spread N/A N/A 0.9 (0.53) 2.0 (0.44) -0.0 (0.38)
A,x spread N/A N/A -28.8 (0.65) -32.5 (0.54) -36.0 (0.47)
azxspread N/A N/A 4.4 (1.69) 5.1 (1.39) 5.4 (1.21)
pgxspreadN/A N/A 0.3 (0.31) 0.1 (0.25) 0.0 (0.22)
fcxspread N/A N/A 19.0 (0.17) 19.4 (0.14) 19.4 (0.12)
,uz 11.7 (0.05) 12.5 (0.06) 13.5 (0.30) 14.1 (0.25) 13.9 (0.23)
p; N/A -12.0 (1.03) N/A -17.0 (0.85) -14.9 (0.80)
Oxyz N/A N/A 5.7 (0.35) 7.2 (0.29) 7.0 (0.25)
A,xpz N/A N/A -20.8 (0.44) -21.4 (0.37) -20.9 (0.32)
0'2po N/A N/A -9.5 (1.14) -9.6 (0.95) -9.3 (0.82)
p}, xpz N/A N/A -0.0 (0.21) -0.0 (0.17) 0.0 (0.15)
fcxpz N/A N/A 12.4 (0.12) 13.1 (0.10) 13.1 (0.08)
spread xpz N/A N/A N/A N/A -1.3 (1.28)

R2 0.763 0.814 0.819 0.876 0.907

 

data size: 91,101
standard error in parenthesis

41

 

 

Table

 

constant

9 -
19: N
.1. -«
.13 \
a; N
n; -<
pf N

fitted costs 1

fc' N‘
9x A, N’
01 0.7 N
9*- p; .\'
9x fc N‘
.i,x 0:, N‘
.~l,x p; \"
14.x fc N‘
0” ”’1 N I
0; X fc \
P1X fc N‘ ..
SDread 7

Spread: N
9x Spread \',‘
‘l-’ X Spread N7;
0'5 X Spread N ”,1
L)‘ X Spread \',',1
1c X Spread N ”,1
“1 19,14
1:; N1

 

 

Table 3.11 Measure of Minimum Disinvestment (Natural Logged), Ln 0‘

Linear with Quadratic with Quadratic with

 

 

Linear Quadratic Interaction Interaction (1) Interaction (2)

constant 4.05 (0.02) -1.01 (0.00) -0.59 (0.01) -1.00 (0.01) -0.18 (0.01)
(9 -7.37 (0.02) 2.11 (0.01) 1.18 (0.01) 2.06 (0.01) -0.33 (0.02)
192 N/A -l.01 (0.01) N/A -1.06 (0.01) 0.05 (0.01)
A, -6.22 (0.03) 2.45 (0.01) 0.49 (0.01) 2.26 (0.01) -1.00 (0.02)
A: N/A -1.79 (0.02) N/A -1.84 (0.01) 0.36 (0.02)
oz 1.24 (0.07) -0. 14 (0.04) -1.20 (0.02) -0.46 (0.03) -0.53 (0.03)
0; N/A -6.74 (0.32) N/A -6.42 (0.26) -6.42 (0.15)
pf, -0.01 (0.01) -0.44 (0.01) -0.32 (0.00) -O.44 (0.01) -0.34 (0.01)

p;2 N/A 0.12 (0.01) N/A 0.12 (0.01) 0.11 (0.00)
fixed costs 1.96 (0.01) -1.17 (0.00) -0.41 (0.00) -0.94 (0.00) -l .44 (0.00)
fc2 N/A 0.49 (0.00) N/A 0.48 (0.00) 0.49 (0.00)
0x A, N/A N/A N/A N/A 3.12 (0.02)
6x oz N/A N/A N/A N/A 0.32 (0.03)
19 x p; N/A N/A N/A N/A -0.50 (0.01)
6x fc N/A N/A N/A N/A 1.08 (0.00)
A, x oz N/A N/A N/A N/A 0.15 (0.04)
A, x pf, N/A N/A N/A N/A 0.30 (0.01)
A, x fc N/A N/A N/A N/A 0.58 (0.00)
0'2 x p], N/A N/A N/A N/A -0.13 (0.02)
oz x fc N/A N/A N/A N/A -0.03 (0.01)
p], x fc N/A N/A N/A N/A -0.14 (0.00)
spread 22.40 (0.09) -4.34 (0.04) -4.55 (0.09) -4.94 (0.07) -6.66 (0.04)
spread2 N/A 2.42 (0.42) N/A 0.29 (0.35) 6.49 (0.21)
0x spread N/A N/A 2.60 (0.11) 2.57 (0.08) 4.05 (0.04)
A, x spread N/A N/A 3.76 (0.13) 4.45 (0.10) 4.82 (0.05)
02 x spread N/A N/A 0.01 (0.34) 0.17 (0.25) 0.28 (0.14)
p], x spreadN/A N/A 0.51 (0.06) 0.56 (0.04) 0.41 (0.03)
fc x spread N/A N/A -4.20 (0.03) -4.18 (0.02) -3.91 (0.01)

g; 19.00 (0.06) -0.76 (0.01) -0.82 (0.06) -0.33 (0.04) -1.56 (0.03)
,1; N/A -9.84 (0.18) N/A -9.22 (0.15) -5.54 (0.09)
19 x #2 N/A N/A -1.79 (0.07) -2.02 (0.05) -1.60 (0.03)
A, x #2 N/A N/A 1.25 (0.09) 1.35 (0.06) 1.43 (0.04)
oz x ,uz N/A N/A 11.70 (0.23) 10.80 (0.17) 0.39 (0.19)
p; x #2 N/A N/A -1.27 (0.04) -1.31 (0.03) -1.33 (0.02)

fc x #2 N/A N/A -0.72 (0.02) -0.85 (0.02) -0.81 (0.01)

spread xyz N/A N/A N/A N/A 16.60 (0.15)

R7 0.779 0.927 0.906 0.951 0.984

 

data size: 91,101
standard error in parenthesis

42

 

 

 

Table 3.12

t?

61:

.11,

.15

0':

p.1-

1: 3
fixed C0515

fc: )
9x .4 1
9X 0:? .\
9* p; 31
9x fc T\
:1,x O’ .\
.1, x p; N
.1, x fc \-
‘72 X If;- x
0: X 1c .\'
Pi; X fc .\'
Spread 9

5P1 63d: N
9“ Spread N
‘4“ Spread N
0” "‘ Spread N
p1 x Spread N71
ya x Spread N”:

I“).

Table 3.12 Measure of Inaction Range (without Fixed Costs, Natural Logged), Ln G

Linear with Quadratic with Quadratic with

 

 

Linear Quadratic Interaction Interaction (1) Interaction (2)
constant 5.3 (0.01) 9.4 (0.03) 5.1 (0.03) 9.4 (0.03) 1.8 (0.07)
19 -0.9 (0.02) -7.7 (0.06) -0.5 (0.04) -7.8 (0.05) 7.9 (0.15)
62 N/A 8.4 (0.07) N/A 8.8 (0.06) 3.5 (0.09)
A, -3.2 (0.02) -l4.6 (0.10) -0.8 (0.05) -127 (0.09) 5.0 (0.21)
A; N/A 11.4 (0.11) N/A 12.1 (0.09) 1.7 (0.17)
a, 1.8 (0.01) 0.2 (0.24) 2.3 (0.13) 0.7 (0.22) 0.9 (0.30)
0; N/A 14.7 (2.10) N/A 14.2 (1.73) 14.1 (1.36)
p; -3.2 (0.01) -6.5 (0.07) -3.6 (0.02) -6.8 (0.06) -4.2 (0.06)
pg? N/A 3.2 (0.07) N/A 3.2 (0.06) 3.3 (0.04)
fixed costs 2.8 (0.01) 4.2 (0.02) 1.3 (0.01) 2.6 (0.02) 6.0 (0.03)
11:2 N/A -13 (0.02) N/A -1.2 (0.02) -1.2 (0.01)
am, N/A N/A N/A N/A -16.1 (0.23)
19x 02 N/A N/A N/A N/A -0.5 (0.29)
6x1); N/A N/A N/A N/A -6.8 (0.05)
0x fc N/A N/A N/A N/A -4.2 (0.03)
A,x 02 N/A N/A N/A N/A -0.8 (0.37)
A,x ,9, N/A N/A N/A N/A -2.2 (0.07)
A,x fc N/A N/A N/A N/A -5.7 (0.04)
02x p, N/A N/A N/A N/A -0.1 (0.17)
a, x fc N/A N/A N/A N/A 0.5 (0.10)
p; x fc N/A N/A N/A N/A 0.4 (0.02)
spread 9.6 (0.09) 3.9 (0.27) 10.9 (0.57) 4.4 (0.49) 8.0 (0.39)
spread2 N/A 70.7 (2.79) N/A 86.7 (2.30) 85.4 (1.88)
axspread N/A N/A -7.8 (0.67) -5.8 (0.51) -9.4 (0.41)
A,xspread N/A N/A -36.7 (0.82) -40.2 (0.63) -44.2 (0.50)
azxspread N/A N/A 5.1 (2.13) 6.0 (1.64) 6.6 (1.30)
pgxspreadN/A N/A 7.6 (0.39) 7.1 (0.30) 6.3 (0.23)
fcxspread N/A N/A 24.8 (0.22) 25.7 (0.17) 25.3 (0.13)
#2 12.9 (0.06) 14.1 (0.07) 16.5 (0.38) 16.5 (0.30) 16.7 (0.25)
,1; N/A -12.4 (1.21) N/A -17.8 (1.00) -15.7 (0.85)
6po N/A N/A 5.4 (0.44) 7.7 (0.34) 6.6 (0.27)
21,po N/A N/A -220 (0.56) -22.2 (0.43) -21.9 (0.34)
6,po N/A N/A -249 (1.45) -249 (1.11) -24.2 (0.88)
pgxyz N/A N/A 0.3 (0.26) 0.5 (0.20) 0.7 (0.16)
fcxyz N/A N/A 11.5 (0.15) 13.0 (0.11) 13.1 (0.09)
spread x ,1, N/A N/A N/A N/A -4.8 (1.37)

R2 0.783 0.849 0.826 0.898 0.936

 

data size: 91,101
standard error in parenthesis

43

 

 

4.1, Econon
This chaptcr i
Costl)“ reversil

theoretical mt

and

Both axes in 111

5126 does not at
dlagram becom

parallel. and II

lL‘

CHAPTER 4

ECONOMETRIC PROCEDURE

 

 

4-1. Econometric Model 1
This chapter investigates actual investment in accordance with the theoretical model of l
costly reversible investment with fixed costs. The optimal investment rule of the
theoretical model is
(1) K,, = Z,/y.;,, if Z, > 2*(= y;,K,_), (38-a) by
(O) K,,th, ifZ'SZ,_<_Z+, (38-b)
and (D) K,, = Z,/y;.,, if z, < Z‘(= y;,1<,__). (38-c)

Both axes in figure 3.1 are normalized by pre-investment capital stock, K,_, so that firm’s
size does not affect the analysis. After both axes are natural-logged, the schematic
diagram becomes figure 4.1. In figure 4.1, investment locus and disinvestment locus are

parallel, and their slope is forty five degree.

 

 

I
I
LnG' —————— | :
I I
I I
I I
1

(01 1 (I)
Lnyfrr Lnyfrr Ln (Zn/Kc-l

Figure 4.1 - Normalized Investment Function

44

 

This

 

K»). {Cli’resc
represented

for 311315.515.

By USlng [hC

and G : l1"

(O) k-:0

(D) k: = L

The theoretical
are functions 01

variable. Th C)’ 1

L

L1

L11

Here. :, is a vecto

constant). This a1

 

otehastic variablt
Intestment and t

11‘
M00168 ( ’7( = 3p;

 

This analysis uses annual data. The capital stock at the beginning of the period,

KM, represents post-investment capital stock, K,+. Pre-investment capital stock, K,-, is

represented by 12'" K,. By assuming a depreciation rate ( 5 )_, the measure of investment

for analysis, k,, is

k, = Ln(KH /K,_ ) = Ln(Km /K, )+ a . (39)
By using the ratios of the critical values as G + = y}; /y,+.u ( > 1 ), G" = y}, /y,70 (< l ),
and G = y; /y.,‘., (> 1 ), the optimal investment rule becomes as follows:

(I) k, = -Ln y}, + Ln(Z, /K, )+ 5
= LnG+ — Ln y}, + Ln(Z, /K, )+ 5, if Ln(Z, /K, )+ 5 > Ln y], (40-a)

(0) k, = 0, if Ln y;,(= Ln y; — LnG): Ln(Z, /K, )+ a s Ln y; (40-b)

(D) k, = LnG‘ — Ln y;, + LnG + Ln(Z, /K, )+ a,
if Ln(Z, /K, )+ 5 < Ln y; — LnG (40-C)

The theoretical model shows that the logarithms of the four values, y; , G+ , G" , and G

are functions of economic parameters such as the spread and trend of the stochastic

variable. They can be approximated by linear equations.

Ln y}; =fi( spread, pg, 6, A,, 0'2, fixed costs, pg. ) z z,v (41-a)
LnG” =f2( spread, ,uz, 6, A,, 02, fixed costs, pf, ) z z,v; (41-b)
Ln G ' =f3( spread, pg, 6, A,, 0“,, fixed costs, pg. ) z z,v; (4l-c)
LnG =fi( spread, pg, 6, A,, 0'2, fixed costs, pg. ) z z,vg (4l-d)

Here, 2, is a vector of parameters, where z, E (spread, pg, 02, 6, A,, fixed costs, pg ,

constant). This analysis assumes that the depreciation rate (6), trend and volatility of the

stochastic variable ([12 and 02), the operating profit function (A,r Z,"0K,9 ), fixed costs of

investment, and the resale price of capital ( p; ) are all constant over time. Therefore, 2,

becomes ( y( = spread + #2 ), constant).

45

 

 

Ther
and technolo
tinn's output
technology is
coefficients.
assuming frict
Re. is the firm
01 l - 1’5) 31
degree one in 1

In addit
and Variance 0

Also. the econo

possible heteror

yr. 5 Ln( Zr
:1 Ln([1
:1 Ln(k

ISL“);

 

 

 

 

 

 

There are two additional assumptions about the market demand of a firm’s output

and technology to measure the economic indicator variable, Z,. The market demand of
firm’s output is assumed to be iso-elastic, i.e., Q, = (1)/Xl )5 ( 8> 1 ), and, the

technology is Cobb-Douglas, i.e., Q = X 2K “L” . Both X . and X, are stochastic
coefficients. Then, the stochastic variable, Z,, is the product of X l and X2, and, by

. . . . . -a _ . (l—at— I,
assummg frlctlonless adjustment of flow mputs, Z, = [Re,K, (wL),/i" T/ l ). Here,

Re, is the firm’s revenue ( = PQ ), (wL), is the cost of flow inputs, and a, and )6, are

a( 1 — 1/6‘) and ,6 (1 — 1/5) , respectively. The stochastic variable, Z,, is homogeneous of
degree one in Re,, K,, and (wL), under these assumptions.

In addition, there is an error, u,, which is assumed to be normal with zero mean
and variance 0'2 , and independent and identically distributed, i.e., u, ~ i.i.d. N[0, 0'2 ].
Also, the econometric model includes company dummy variables which represent

possible heterogeneity among firms. By defining

y, 5 Ln( Z, /K, ) + dummy variables + u,

=(Ln(Re.),Ln(K1), Ln((wL).))( 1 / VI, -( 1 - ,3: )/ W, - fl./ VI)’ + dummies +u,
= ( Ln(Re,), Ln(K,), Ln((wL),), d1, d2, d3, ) 77 + u, = x, 7] + u,,

fl 5 Ln y; - 6w 2“,, V,, 21.1.6.5 ( 7), and vfs (coefficient for y),

the econometric model becomes the following.
( 1 ) k. = 21V; + 16.77 - anvf + 14,, if x,77 + u, > z,,,,,.v_, (42-a)

(O) k,=0, ifz v —z,vg5x,77+u,Sz v ' (42-b)

l,nm.' [ IJmc f

(D) k, = z,v; +x,77—z,.,,,,.v, +z,v, +u,, if x,77+u, <z v, —z v (42-c)

I .Hm' I g

The dummy variables absorb the constant term in z,v,- (= Ln y}, —c§), since y: and Ln y},

always appear in a pair and we cannot identify their constant terms separately. Then, this
analysis uses a generalized version of the fiction model (Maddala 1983, pp. 162). The
likelihood function of the model is the following.

46

+ _
Lln,v,,v,.,v,,v,,0'1k,,x,,z,J

= “lg/{1" — {3 V;}—x,77+ Z,.,,,,,v,]
, a

 

 

 

 

0'
z,.,,,,,.v, — x,17 z,‘,,,,.v, — {2,12, } — x, 77 (43)
(I) -(I)
4111 . l t - 11
1 k, — {z,v; }—x,77+ z,.,,,,,.v», —z,v,.
471.11 .

Here, (15 and (I) are the probability density function (p.d.f.) and the cumulative distribution
function (c.d.f) of the standard normal distribution, respectively. Terms in braces are
additions to the friction model. In other words. some explanatory variables. i.e., 2,. have
different coefficients in different places.
4-2 Analysis Methods
i. Data Sources
The financial data that this analysis uses are the sales revenue (Re,), the book value of
capital stock (K,), and the cost of goods sold ((wL),) in Compustat PC Plus. Although
value added and labor costs could be appropriate for Re, and (wL),, such data is not
available in Compustat PC. Compustat PC Plus contains corporate financial data over 20
years from 1977 to 1996. It has the beginning balance of capital stock as well as the end
balance of capital stock. Therefore, we have 21 years data of capital stock from 1977
tol997. This analysis investigates three industries: the computer and office equipment
industry (SIC 3570), the automobile industry (SIC 3711), and the airline industry (SIC
4512). Three or four companies were selected from each industry. The choice of the
industry is arbitrary, while the choice of the companies is based upon the observations
that some companies had decreases in their capital stock during the examined period. and
that each company has a complete set of data, i.e., the panel data is balanced.

The economic data that this analysis employs are the producers price index (PPI),

the consumers price index (CPI) and the federal funds rate. The PPI is used as the

47

numeraire. The discount rate or a risk free interest rate ( y) is the difference between the
federal funds rate and the CPI rate of inflation.
ii. Procedure
(a) Estimation and Measure of Zero Investment
In order to identify firm’s annual capital investment as positive, zero or negative, the
depreciation rate, 6, and its range, Ab‘, are assumed. Then, investment is classified as
positive, zero or negative by the ratio of capital stock between two periods. The ratio of
capital stock is calculated as either
(a) k, = Ln( K,,, / K,.,,_. ) + 6, (44a)

or (b) k, = Ln( K,,, / K, ) / n + 6. (44b)

For example, when the ratio of capital stock is greater than +A5, then investment at time
t is classified as positive investment. Similarly, if the ratio of capital stocks is between
—A5 and +135, or less than —A6 , investment at time t is classified as zero or negative,
accordingly. Then, the analysis uses equation (43) for the maximum likelihood
estimation.
(b) Test of Model Selection
The analysis compares five investment models by the likelihood of the corresponding
econometric models: (1) reversible investment without fixed costs estimated by OLS, (2)
irreversible investment without fixed costs estimated by a censored Tobit model, (3)
irreversible investment with fixed costs estimated by a generalized version of the Tobit
model (G Tobit), (4) costly reversible investment without fixed costs estimated by a
ristricted version of the friction model (RPM), and (5) costly reversible investment with
fixed costs estimated by a generalized version of the friction model (GF M). By
comparing the likelihood of each model, the analysis selects the best model according to
the selection criteria of Voung (1989). Table 4.1 shows these models, the corresponding

econometric methods, and their likelihood functions.

48

Table 4.1 Investment Models and Econometric Methods

 

Capital Investment

Econometric Method and Likelihood Function

 

Reversible w/o fixed costs

 

 

Ordinary Least Squares (OLS)

L, =Hi¢[k' —x,n+z,.mv,]
a or

 

 

Irreversible w/o fixed costs
K e 1

71

 

o gfloim‘mfllL 2,
2+

 

Censored Tobit (Tobit)
L2 = 1114,“ — xm + amt/f)
l a a

 

x n[1 - <11 x’" ' 2”“ V’ j]
0 0'

 

Irreversible with fixed costs
K t +

 

o , HQL L511- 2t

Generalized Tobit (G Tobit)
L3 = l'I _1_¢[k, —x,r7+ zwcvf - z,v,J

 

l 0'

 

 

 

Restricted Friction Model (RF M)
L. =Hi¢(k' —x,77 + amt/f]

 

I 0’ a

X I—I[(D[xl'7 — z!.nocvf + zlvg )- ¢[xln — 2:310ch ]:l
0 0' 0'

x [11¢[k' — x,n + zwcvf — 2':ng

 

 

 

00' a

 

 

Kt+

 

 

Generalized Friction Model (GFM)

L, =ni¢[k’ —x,17+ 2,,fo —z,v;]
I a a

x n[¢[x,n — 2,,mv, + z,v, ]_ ¢(x,77 — zwcv, j]
0 a a

r -x:77 + 2!.»ch _Ztvg 'ZrVé]

 

 

 

 

0'

l k

 

49

 

Voung (1989) proves that the logarithm of the ratio of likelihood, LR( - ),

converges to the chi-squared distribution when the two competing models:

a: 00

models are nested, the classic Neyman-Pearson LR test is valid. However, if the two

x;6);6 e 8} and G, = {g0} x; y); y e F}, are nested. In other words, when two

 

 

models are non-nested, the ratio converges to a normal distribution. Under the null

hypothesis that two models are equivalent,

1)
2 LR( - )—> 1,2,- when two models are nested, and

q ’
, I)
n""2 LR( - )/ a} —>N[ 0, 1 ], when two models are non-nested.

A
!

Here, LR(6 )7): ZLnlf(Y,'X,;6)/g(Y,|X,;}9)J and

(2)2 = (Mn): {Ln[f(1’,lX,;6)/g(1’,|X,;)7)]}2 —{(1/n)ZLn[f(Y,

the classical LR test is a special case in Voung’s test of model selection.4

X,;6)/g(Y,|X,;;7)]}2. Thus,

 

(c) Test of Serial Correlation

This analysis examines whether residuals, 2?, , are serially correlated. The test is based
upon Wooldridge (1995, pp351). The null hypothesis is that there is no serial correlation.
This test uses three types of residuals. Two are residuals from OLS and residuals from
the GF M. Another is residuals from a two step estimation (TSE) for a limited dependent
variable model.5 The reason that this analysis includes TSE residuals is that the test of
serial correlation in this analysis is based upon OLS residuals and the second step of the

TSE is OLS. The procedure is two steps.

lst step: regress the lagged OLS residuals or the lagged GFM residuals, 1? on

1.1-1 9

x, and z,,, or regress the lagged TSE residuals on x,,, z, and the estimated

inverse Mills ratio. Then, save the residuals, f.

I,I—l ‘

 

‘ Appendix D discusses the test of model selection under the serially correlated error
assumption.
5 Appendix E shows the TSE and its results.

50

2nd step: ( under homoskedasticity assumption ) regress 12,, on r, The t-

.I—I '

statistics on 0.1—1 is asymptotically normal under the null hypothesis.

( under heteroskedasticity assumption) regress 1 on 17,, xr', Then,

.I—l '
N i

2(7). — 1) — SSR is asymptotically chi-squared of degree one under the null
i=1

hypothesis.

(d) Determination of Depreciation Rate, 6

In order to estimate the depreciation rate, 6, the analysis repeats the procedure with

different values of the assumed depreciation rate and its range. The estimate of the

depreciation rate is determined by four criteria: ( 1 ) high likelihood, ( 2 ) homogeneity

of degree one for Z,, ( 3 ) the signs of the estimated critical ratios, i.e., Ln G+ >0,

LnG‘ <0, and Ln G >0, and ( 4 ) superiority of the GFM.

(e) Testing Hypotheses

a. ( Ho ) 77R,.+ 7711+ 77..., = 0; this is a test of whether the ratio, Z, / K,, is homogeneous of
degree zero in the firm’s revenue, capital stock and costs of flow inputs. ( Wald test )
This substitutes for a test of whether Z, is homogeneous of degree one in the three
variables.

b. ( Ho ) two competing investment models are equivalent vs ( Ha ) one model is

better than the other. ( Voung’s Test of Model Selection )

51

CHAPTER 5

INDUSTRY ANALYSIS

This chapter examines three industries: the computer and the office equipment industry
(SIC 3570), the automobile industry (SIC 3711), and the airline industry (SIC 4511), in
accordance with the procedure in chapter 4. The analysis chooses three or four firms
from each industry. The analysis confirms that actual investment is compatible with the
theoretical analysis.

5-1. Computer and Office Equipment Industry

i. Estimation and Measure of Zero Investment

Table 5.1 shows the measure of zero investment for the computer and office equipment
industry. According to the four criteria, the case with net capital stock, 6 = —0.005,

A6 = 0.01,, and n = l is most compatible with the economic model, although cases with 6
close to zero show very similar results.6 Then, the measurement of investment for this
analysis is Ln( K,,. / K, ) - 0.005. The case has a high likelihood, the sum of the
estimated coefficients for the three financial variables is not significantly different from

zero, the signs of the critical ratios are appropriate, and the OF M has the highest

 

6 There are two issues with a negative depreciation rate, 6. One is that the depreciation
rate used for net capital stock is the accounting rate, rather than the physical or economic
depreciation rate. In the cases with net capital stock, 6 represents the difference between
the accounting depreciation rate and the economic equivalent. The accounting
depreciation rate could be larger than the economic rate. Another issue is that of the
price of capital. The book value of capital depends upon the acquiring costs of capital, or
the purchase price of capital. Even if a firm buys the same capital goods, their prices
vary from time to time. In general, old capital goods are cheaper than new ones due to
the inflation. Consequently, the book value of capital stock could underestimate older
capital, so that the estimated depreciation rate can be negative.

52

.688 2:888 05 ~33 0338800 ”20m
The» $3 =8§m-§oz 33820 05 E “50$:me 2a Eon—802: .«o 380 Bfibemunfii
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55

likelihood in the five competing models. When the analysis uses gross capital stock, the

sum of the estimated coefficientS, 77).... 77K and 77...), seems to be significantly different from

ZCI‘O .

Table 5.2 shows the estimated coefficients of the best case with net capital stock,

6 = —0.005, A6: 0.01 and n = l. The sum of the estimated coefficients, 77"., 77K and 77...),

is —0.019, which is very close to zero. The log likelihood of the GFM exceeds the log

likelihood of the next best model, which is OLS, by a margin of about nine. Fixed costs

of disinvestment seem larger than fixed costs of investment, since the absolute value of

the estimated constant for v; is about three times greater than the estimated constant for

v; for the GFM. Those estimated constants are —O.616 and 0.225, respectively.

Table 5.2 Estimation of Computer and Office Equipment Industry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OLS Tobit G Tobit RFM GFM
g... O.671(O.12) 0.642(0.13) 0.493(0.13) O.698(O.l3) 0.559(0.13)
27,. -0.386(0.05) -O.371(0.06) -O.327(0.06) -0.396(0.06) -O.351(0.06)
m1. -0.309(0.07) -O.297(0.08) -O.190(0.08) -O.325(0.08) -0.227(0.07)
d1 (DEC) -0.461(0.30) -o.405(0.30) -O.730(63.3) -o.499(0.31) -0.591(116)
d2 (HWP) -O.478(0.30) -O.426(0.31) -0.774(63.3) -o.513(0.32) -0.635(116)
d3 (IBM) -0.407(0.34) -O.358(0.34) -0.660(63.3) -0.446(0.35) -0.531(1 16)
V,:y -0.019(0.01) -0.019(0.01) -0.010(15.4) -0.020(0.01) 0.040(31.6)
V,: 7 N/A N/A N/A 0.001(0.01) 0.005(46.2)
12,: _cons N/A N/A N/A 0.015(0.02) 0.764(161)
V; : y N/A N/A 0.006(15.4) N/A 0.057(31.6)
15;: _cons N/A N/A O.480(63.3) N/A 0.225(1 16)
.4: y N/A N/A N/A N/A 0.026(33.8)
.4: _cons N/A N/A N/A N/A -O.616(1 12)
a 0.086(0.01) 0.078(0.01) 0.068(0.01) 0.088(0.01) 0.071(0.01)
Pr[2n=0] 0.345 0.272 0.250 0.372 0.378
Ln L, 62.397 45.693 59.658 52.502 71.366

 

 

 

 

 

 

Standard errors in parentheses

Parameters: net capital stock, 6: —0.005, A6: 0.01, and n = 1
Number of observations: total 60, investment 47, zero investment 2, and disinvestment ll

 

ii. Test of Model Selection

Table 5.3 shows the statistics that Voung (1989) proposed for model selection. Two
cases: (a) Tobit vs G Tobit and (b) the RFM vs the GFM, are nested so that the classical
LR test is valid. All other cases are non-nested. According Voung’s test of model
selection, the GF M is significantly better than any other model. When the test compares
the GFM with any other model, Voung’s statistics for the GF M is significant and
positive. By comparing the Tobit model with the G Tobit model and the RF M with the
GF M, we observe significant and positive values in the statistics, 27.930 and 37.728,
respectively, so that we conclude that fixed costs are significantly different from zero. At
the same time, by comparing the G Tobit model with the GFM, we also observe
significant and positive values in the statistics, 2.446, so that incorporating disinvestment
has a significant effect on the analysis of investment.

In addition, OLS is significantly better than the Tobit model and the RF M, since
Voung’s statistics is significant and negative for two cases. Therefore, the tests suggest
that a partial specification of investment model, e.g., costly reversible investment or
irreversible investment without fixed costs, is not appropriate to analyze actual

investment behavior.

Table 5.3 Model Selection for Computer and Office Equipment Industry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F g \ G, OLS Tobit G Tobit RFM GFM
OLS 0.9333 0.6808 0.5173 0.2483
Tobit -2.232*** 0.3748 1.0034 0.9351
G Tobit -0.429 27.930*** 1.1674 0.3820
RFM -1.776* 0.878 -0.855 0.7068
GFM 2.324“ 3.427*** 2.446*** 37.728***

Lower left: \éoung’s statistics, (NT)"” LR( - )/ a“) or 2 LR( - ); a positive value favors F 0
over ,.

Upper right: sample variances of the log-likelihood ratio, a32
Parameters: net capital stock, 6: —0.005, A5: 0.01, and n = 1
*: reject H0 at 10 % **z reject Ho at 5 % ***: reject H0 at 1 %

57

iii. Test of Serial Correlation

Table 5.4 shows the test of serial correlation for the computer and office equipment
industry. The test with the TSE residuals and the GF M residuals fails to reject the no-
serial-correlation hypothesis, while the test with the OLS residuals rejects the hypothesis
at the five percent level. Therefore, the test of serial correlation also favors the GFM.
The TSE residuals and the GFM residuals yield essentially the same probability for no
serial correlation. In addition, the homoskedasticity case and the heteroskedasticity case

yield very close probabilities for no serial correlation.

Table 5.4 Serial Correlation Test for Computer and Office Equipment Industry

 

 

 

V OLS Residuals TSE Residuals GF M Residuals
Homoskedasticity t-stat. for r",_l = 2.063 t-stat. for f,_, = 0.565 t-stat. for r‘,_l = 0.584
Case Pr[ - ] = 0.044 Pr[ . ] = 0.574 m - ] = 0.562
Heteroskedasticity 2(7)-1)-SSR = 4.027 Z(T,--1)-SSR= 0.324 2(7)-1)-SSR = 0.345
386 Pr[ - ] = 0.045 Pr[ - ] = 0.569 Pr[ - ] = 0.557

 

 

 

 

 

 

Parameters: net capital stock, 5= —0.005, A6: 0.01, and n = 1
Number of data: OLS; 2 (T,- — 1) = 57, TSE and GFM; 2 (T,- — 1) = 53 (the difference is
due to zero investment.)

5-2. Automobile Industry

i. Estimation and Measure of Zero Investment

Table 5.5 shows the measure of zero investment for the automobile industry. The case
with net capital stock, 6 = —0.02, A5 = 0.01, and n = 2 (b), shows the best result according
to the four criteria. Then, the measurement of investment for this analysis is

Ln( K,,; / K, ) / 2 - 0.02. When the analysis uses gross capital stock, the program for

analysis often fails to converge.

58

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61

Table 5.6 is the estimated coefficients of the best case that yields the highest

likelihood. The sum of the estimated coefficients for 77,-,” 77K and 77...), is 0.029, and very

close to zero. And, the log likelihood of the GF M exceeds the log likelihood of the

second best model, OLS, by a margin of about seventeen. Fixed costs of investment and

fixed costs of disinvestment seem to be of comparable size, since the estimated constant

for v; is as large as the absolute value of the estimated constant for v; for the GF M.

Those estimated constants are 0.301 and —0.367, respectively.

Table 5.6 Estimation of Automobile Industry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OLS Tobit G Tobit RFM GFM
27,-- 0.519(0.22) 0.597024) 0.128( 0.19) 0.563(029) 0.163( 0.17)
:7.- 01010.03) 01120.03) -0.138( 0.03) 00990.03) -0.110( 0.02)
71-- 04170.27) 04810.29) 0.065( 0.24) -0.468(0.29) -0.024( 0.21)
d1 (C) 01130.53) 01590.57) -1.060(26.81) -0.062(0.56) -0.621(28.16)
d2(F) -0.146(0.59) -O.206(0.64) -1.180(26.81) -0.088(0.63) -0.687(28.16)
d3(GM) -0.097(0.61) -0.l45(0.66) -1.143(26.81) -0.037(0.65) -0.662(28.16)
11:7 00000.01) 00000.01) -0.024( 5.77) 00000.01) 00000024)
0,: 7 N/A N/A N/A 0002000) 00020559)
v,:_co'ns N/A N/A N/A 0.010(001) 0.490(39.78)
V;: 7 N/A N/A 0015( 5.77) N/A 00070024)
5: _cons N/A N/A 0.457(26.80) N/A 030108.16)
V1 - y N/A N/A N/A N/A 0.014(11.79)
v1”: _cons N/A N/A N/A N/A -0.367(28.09)
0' 0.067(0.01) 0.071001) 0.046( 0.00) 0.071001) 0.046( 0.00)
Pr[)Zn = 0] 0.984 0.954 0.239 0.935 0.507
Ln L,- 73.004 43.434 74.923 57.450 89.875

 

 

 

 

 

 

Standard errors in parentheses

Parameters: net capital stock, 5: —0.02, A6= 0.01, n = 2 (b)
Number of observations: total 57, investment 45, zero investment 3, disinvestment 9

62

 

 

ii. Test of Model Selection
Table 5.7 shows Voung’s test of model selection for the automobile industry. Voung’s
test of model selection ranks the five models as follows:

GFM > OLS z G Tobit > RFM > Tobit.
For the automobile industry, the generalized Tobit model is as good as OLS. Also, there
are significant fixed costs of investment and disinvestment, since the GFM and the G
Tobit model are better than the RPM and the Tobit model, respectively. And, because the
GFM and the RF M are better than the G Tobit model and the Tobit model, respectively,

disinvestment is important for analysis.

Table 5.7 Model Selection for Automobile Industry

 

 

 

 

 

 

 

 

 

 

 

 

 

F 9 \ G, OLS Tobit G Tobit RFM GFM
OLS 0.8998 0.5576 0.8937 0.8283
Tobit -4.129*** 0.6821 0.7782 1.9468
G Tobit 0.340 62.978*** 0.7603 0.5120
RFM -2.179*** 2.104“ -2.654*** 0.9630
GFM 2.455*** 4.409*** 2.768*** 64.850***

Lower lefi: \éoung’s statistics, (1)/T)" 2 LR( - ) la“) or 2 LR( - ); a positive value favors F 9
over ,.

Upper right: sample variances of the log-likelihood ratio, (:32
Parameters: net capital stock, 5: —0.02, A6: 0.01, and n = 2 (b)
**z reject H0 at 5 % ***: reject H0 at 1 %

iii. Test of Serial Correlation

Table 5.8 shows the test of serial correlation for the automobile industry. The TSE
residuals fail to reject the no-serial-correlation hypothesis at the ten percent level. The
GFM residuals fail to reject the hypothesis at the six percent level. The OLS residuals
reject the hypothesis. The homoskedastic case and the heteroskedastic case yield

essentially the same probability for no serial correlation, similar to the other industries.

63

 

 

 

Table 5.8 Serial Correlation Test for Automobile Industry

 

 

 

 

OLS Residuals TSE Residuals GF M Residuals
Homoskedasticity t-stat. for f,_, = 2.648 g-stat. for f,_, = 1.675 t-stat. for f,_, = 1.902
Case Pr[-]=0.011 Pr[-]=0.101 Pr[-]=0.063
eteroskedasticity 2(T,-1)-SSR = 6.309 (T,-1)-SSR= 2.703 2(7)-l)-SSR = 3.430
386 r[-]=0.012 Pr[-]=0.1002 Pr[-]=0.064

 

 

 

 

 

Parameters: net capital stock, 6 = -0.02, A6 = 0.01, and n = 2 (b)
Number of data: OLS; 2 (T,- — 1) = 54, TSE and GFM; 2 (T, — 1) = 48 (the difference is
due to zero investment.)

5-3. Airline Industry

Since there is an organized international used airplane market, it is easy for airline
companies to resell their used planes, so the industry may have small fixed costs of
disinvestment. The industry experienced a deregulation in the 1980’s. The Airline
Deregulation Act of 1978 initiated this deregulation. Thus, this analysis includes a
deregulation dmnmy variable, dereg (1 for years 21987, 0 for years <1987) in order to
incorporate industry’s adjustments to the deregulation. Without this dummy variable, the
econometric program that this analysis uses often fails to converge. The airline industry,
however, shows results similar to the two industries and that there are significant fixed
costs of investment and disinvestment.

i. Estimation and Measure of Zero Investment

Table 5.9 shows the measure of zero investment for the airline industry. The case with
net capital stock, 5 = —0.05, A6 = 0.01, and n = 2 (b), shows the best result according to
the four criteria. Thus, the measurement of investment for this analysis is

Ln( KM / K, ) / 2 - 0.05. When the analysis uses the gross capital stock, the sum of the
estimated coefficients for 77m, 77,, and 77,), is significantly different from zero, similar to

the computer and office equipment industry.

64

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69

Table 5.10 shows the estimated coefficients of the best case with net capital stock,

6 = —0.05, A6= 0.01, and n = 2 (b). The sum of the estimated coefficients for 77R,” 77K and

 

77...), is —0.026, and very close to zero. And, the log likelihood of the GFM exceeds the
log likelihood of the second best model, OLS, by a margin of about sixteen. Fixed costs

of investment and fixed costs of disinvestment seem to be of comparable size, similar to

the automobile industry. Those estimated coefficients for v; and v; are 0.417 and

—0.389, respectively.

Table 5.10 Estimation of Airline Industry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

OLS Tobit G Tobit RFM GFM

m... -0.016(O.16) -0.047(O.18) 0.277( 0.16) -0.040(O.16) 0.245( 0.13)
m: -0.108(0.05) -0.122(0.06) -0.099( 0.05) -0.108(0.06) -0.096( 0.04)
27..., -0.016(0.12) 0.013(0.14) -O.167( 0.13) -0.002(0.13) -0.175( 0.10)
d1(AMR) 1.238(0.37) 1.411(0.44) -O.548(48.54) 1.373(039) -0.112(31.23)
d2(DAL) 1.191(0.36) 1.311(0.43) -O.603(48.54) 1.278(O.38) -0.172(31.23)
d3(UAL) 1.218(0.39) 1.351(045) -0.605(48.54) 1.308(0.40) -O.163(31.23)
d4 (U) 1.151(0.33) 1.261(0.39) -O.474(48.54) 1.228(0.35) -0.100(31.23)
dereg 0.046(0.04) 0.059(005) 0.003( 0.05) 0.048(0.05) 0.025( 0.04)
.,,:y -0.008(0.00) -0.010(0.01) 0.002(20.26) -0.008(0.00) 0.001( 8.72)
V1: y N/A N/A N/A 0.002(000) 0.006(20.20)
vi:_cons N/A N/A N/A 0.012(0.01) 0.673(55.16)
.4: 7 N/A N/A 0.004(2026) N/A 0.004( 8.72)
v;: _cons N/A N/A 0.547(48.59) N/A 0.417(3123)
0;: 7 N/A N/A N/A N/A 0.0190 8.21)
v; _cons N/A N/A N/A N/A -O.389(45.54)
0' 0.083(0.0l) 0.088(0.0l) 0.069( 0.01) 0.086(0.01) 0.064( 0.01)
Pr[En = 0] 0.002 0.003 0.849 0.001 0.498

Ln L,- 81.545 40.702 68.770 66.292 97.547

 

 

 

 

 

 

 

Standard errors in parentheses
Parameters: net capital stock, 6: —0.05, A6: 0.01, n = 2 (b)
Number of observations: total 76, investment 55, zero investment 3, disinvestment 18

70

ii. Test of Model Selection
Table 5.11 shows Voung’s test of model selection for the airline industry. Voung’s test
of model selection ranks the five models as follows:

GFM > OLS > G Tobit z RF M > Tobit.
For the airline industry, the RFM is as good as the G Tobit. Otherwise, the order is the
same as the other industries. There are significant fixed costs of investment and

disinvestment contrary to expectations. Disinvestment is significant.

Table 5.11 Model Selection for Airline Industry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F 9 \ G, OLS Tobit G Tobit RFM GFM
OLS 0.7572 0.6010 0.6515 0.4352
Tobit -5.384*** 0.2406 0.7450 1.1552
G Tobit -1.890* 56.135*** 0.8555 0.6732
RFM -2. 168* * 3.40.1 * * * -0.307 0.6415
GFM 2.783*** 6.067*** 4023*" 6252*“

Lower lefi: \éoung’s statistics, (NT)*'AZLR( - )/a‘) or 2 LR( - ); a positive value favors F 9
over r

Upper right: sample variances of the log-likelihood ratio, (0’
Parameters: net capital stock, 5: -0.05, A6: 0.01, and n = 2 (b) y
*: reject Ho at 10 % **: reject H0 at 5 % ***: reject Ho at 1 %

iii. Test of Serial Correlation

Table 5.12 shows the test of serial correlation for the airline industry. The TSE residuals
and the GF M residuals fail to reject the no-serial-correlation hypothesis at the ten percent
level, while the OLS residuals reject the hypothesis. The conclusion is the same as the
other industries. The TSE residuals and the GF M residuals yield essentially the same
probability for no serial correlation. The homoskedastic case and the heteroskedastic

case also yield essentially the same probability for no serial correlation.

7l

 

Table 5.12 Serial Correlation Test for Airline Industry

 

 

 

 

OLS Residuals TSE Residuals GFM Residuals
Homoskedasticity I-stat. for r',_l = 3.200 t-stat. for 73., = 1.138 t-stat. for f,_l = 1.153
Case Pr[ . ] = 0.002 Pr[ - ] = 0.259 Pr[ - ] = 0.253
Heteroskedasticity 2(7H)-SSR = 9.075 Z(T,~-l)—SSR= 1.289 2(T,~-1)-SSR = 1.324
Case Pr[ - ] = 0.003 Pr[ - ] = 0.256 Pr[ - 1 = 0.250

 

 

 

 

 

Parameters: net capital stock, 5: -0.05, A6 = 0.01, and n = 2 (b)

Number of data: OLS; Z (T, - 1) = 72, TSE and GFM; 2 (T,- — 1) = 67 (the difference is
due to zero investment.)

 

5-4. Summary E
The empirical analysis investigates three industries: the computer and office equipment I.
industry(SIC 3570), the automobile industry (SIC 3711), and the airline industry (SIC
4512). Although the choice of the industries is arbitrary, all of the examined industries
show many similarities: the homogeneity of the economic indicator variable in the sales
revenue, capital stock and the cost of flow inputs, the superiority of costly reversible
investment with fixed costs or the corresponding GFM, no serial correlation in the GFM
residuals as well as the TSE residuals, and reversible investment without fixed costs or
corresponding OLS as the second best model. By comparing likelihood of the five
investment models, the analysis concludes that fixed costs are significantly different from
zero, and that incorporating disinvestment has significant effects on analysis. Also, as
OLS is significantly better than the other models except the GF M and, possibly the G
Tobit model, the analysis suggests that partial specification of investment, e.g.,
investment without the costly reversibility or fixed costs, is not appropriate to analyze
actual investment.
However, the estimated coefficients are different among the industries, which
suggests heterogeneity among the industries. In addition, for the airline industry, the

analysis requires the deregulation dummy variable so that the airline deregulation of the

72

1980’s could have affected industry’s investment. For the measure of investment, a net
capital stock shows better results than the gross capital stock. The estimated depreciation

rate is zero or negative for the examined industries, when the analysis uses the net capital

stock.

 

73

CONCLUSIONS

This analysis investigated costly reversible investment with fixed costs. The analysis
applied option pricing theory to incorporate the impact of the possibility of future

investment on current investment. Fixed costs of investment introduced minimum

 

investment, while fixed costs of disinvestment introduced minimum disinvestment. Both If»
fixed costs and the costly reversibility resulted in zero investment as the optimal choice.
When the firm contemplated future investment, its investment was lower than when it did
not.
The analysis also investigated how the economic parameters such as the discount in"

rate affected the critical values such as the target value and the trigger value of
investment. As the theoretical model did not yield closed form solutions, this analysis
resorted numerical solutions. More than 90,000 cases for investment with fixed costs and
more than 30,000 cases for investment without fixed costs were examined, so that the
conclusions were robust. All of the critical values, except the trigger value for
investment with fixed costs, were linear in the parameters, while the target value for

in vestrnent was semi-log linear in the parameters.

The empirical analysis confirmed that actual investment behaved as the
theoretical analysis predicted. The analysis examined three industries: the computer and
office equipment industry, the automobile industry, and the airline industry. Although
each industry had different estimated coefficients, they all showed the similar results: the
homogeneity of the stochastic variable, the superiority of costly reversible investment
with fixed costs or the corresponding GF M, reversible investment without fixed costs or
corresponding OLS as the second best model, and no serial correlation in the GFM
residuals. The analysis confirmed that there were the costly reversibility and fixed costs

in actual investment.

74

 

APPENDICES

75

APPENDIX A

TWO PERIOD MODEL OF COSTLY REVERSIBLE INVESTMENT
WITHOUT FIXED COSTS

This appendix solves the model as a two period model, and shows economic intuition of
the model. The model regards future investment and disinvestment as a call option and a
put option, respectively. The appendix shows that value of future investment as the call
option is equal to the expected value of net gain from the second period investment while
value of future disinvestment as the put option is equal to the expected value of net gain
from future disinvestment. The appendix also shows that how first period capital stock
affects the expectation of the second period gain.

This two period model is a special case of Abel, Dixit, Eberly and Pindyck
(abbreviated ADEP) (1996). They use a general form for the profit function, while this
model uses the same profit function as the main text. There are no fiXed costs of
investment in the appendix. There are three differences between this model and the
model in ADEP (1996): In ADEP (1996) there is no physical depreciation, the purchase
price of new capital is different between two periods,7 and the expected PDV of net cash

flow includes costs of the first period investment.8

 

7 In the two period model, the derivative of the expected PDV of net cash flow will not
reach the (second period) purchase price of capital in many cases. In such cases, the first
period price of capital should be lower than the second period price of capital. This
section assumes conditions under which the derivative of the expected PDV of net cash
flow reaches higher than the second period price of capital. Therefore, the purchase price
of capital can be the same for the two periods.

8 ADEP (1996) adopt the Net Present Value (NPV) rule, i.e., if the (expected) net present
value, or the expected PDV of net cash flow less initial investment costs, is positive, the
firm should invest. One consequence is a difference in the optimal investment rule.
When the NPV rule is used, the derivative of the expected NPV, which is called q°,

76

 

 

A firm makes its first investment decision to choose optimally capital stock for
the first period, Km, for a given level of capital stock, Kn, after the economic indicator at
time t = 0, Z0, is revealed. Or, equivalently, the firm determines either d10 (= K0. — K0.) or
6le0 ( = K0. — K,,). At time t = A1, firm’s capital depreciates by the depreciation rate, 6, to
its pre-investment level of capital stock, K .- = e'm Km). Then, after the economy
reveals Z ., the firm makes its second investment decision to determine its post-investment
capital stock, K ,.. The firm will not make any further investment after t = At. In addition,

there are no fixed costs for both investment and disinvestment. Thus, we can rewrite

equation (3) in the main text as follows:

V(KO+9ZO): max EO[{4KK(§)+ fie-(”figbzi—ods — pkdlo + pli'dDoi

{dl .dn}

+e—Wi‘4nK16:fe—(yfierE—gd7_p;dll +pdeli] (45)

subject to K0, = K,_ + d1O — dD0 , Kn; given, and
K,, = K,_ +dI, —dD, (= e‘M’Km + d1, — dD, ).
A-l. Investment Model
First, we solve the second period investment decision. or the maximization of the second

brace in equation (45). Assuming 7> #2,

max E0[An K: fe‘(’+‘w)'Z:‘0dr—pidl, + p;.dD,] (46)

{(11,111)}
subject to K,, = K,_ + 611' —dD,(= e"‘WK0, + d1, —dD, ), K,_: given.
Defining A, a fry/[(7 +619)—(1—6)(pz we; /2)J, y, a 21 /K,, ,

+

yl E (p;( / A}, )l'il‘”), and y," a (p; / Ay)l"’“"), we can rewrite equation (46) as follows.

 

should equal to zero, i.e., the optimal investment rule is q°=0. When the expected PDV
of net cash flow is used, the derivative of the expected PDV of net cash flow, called q,
should equal the purchase price of capital for investment or the resale price of capital for
disinvestment if the firm invests or disinvests.

77

 

Av _ + _
331375151? Z,I ”K3 —- pKdI, + pKdD, |Z,] (47)

Its solution is

o If Z 1 / K ,_ > yf , then the firm should buy new capital such that yl = yf , or

 

d1 = (Z,/y;‘)—K,_.
o If y,‘ S Z 1 / K ,_ S yf , then the firm should neither buy nor sell capital so that EA“
K 1+ = K ,_. :-
0 If ZI / K 1- < y," , then the firm should sell installed capital such that yI = y,‘ , or .
dD=K,_ —(Z,/y;). L
Here, yf and y,‘ are two critical values for the ratio, Zl / K .., in the second period. ..

Then, after solving the first term in equation (45) and incorporating the above optimal

investment for the second period, we can rewrite equation (45) as follows:

V(K0+9ZO)= AnZI‘J—BK:+At
'1 'I— A
+e-W f A {jaw/<3 +PII'(K1— _K|+ )}dF(Zl|ZO)

IKI— 6
A.
+e—yAI [1:10- {jzil-oKli 'Pk(K1+ _Kl—)}dF(Z||ZO)

— [K.. > K.-1p,:(1<.. - K,_ )+ [K,, < K0_1p;(KO- _ K0,) (48)

 

Zl/yf,ifZI>y1+K1_
subject to K,, = K._ ,if ny._ ZZ, nyK._
Zl/yl-sifyl—Kl— >ZI-

Here, F (Z 1 [Z 0) is the cumulative density function of Z . conditional on 20, Since K,_ is

equal to e’aA'Ko. and appears in the limits of the definite integral in equation (48), the

firm’s choice of capital stock in the first period affects not only the net cash flow for the

78

first period but also the expectation of net cash flow for the second period. This is how
the first period decision and the second period decision are linked.

A-2. Values of Future Investment and Disinvestment

From equation (48), we can derive the value of future disinvestment as the put option,
P(K0,Z0), and the value of future investment as the call option, C(K0,Zo) by comparing
equation (48) with the so-called naive case in which the firm assumes zero second period
investment when it makes its first period investment decision. By rewriting equation

(48), we have the following equation.

 

. A,
V(Ko. , Z0 ) = 4.23;" K5181 + e-WM f 6; Z,‘"”K€.dF(Z. IZO)
fin: -'|—K'~ Av 1—0 (9 — Av 1—0 9 —
+3 I 921 K1+"PKK1+ — ‘5“21 K1—_pKKl- dF(Zl|ZO)
—yAI Av 1-9 a + Av 1—0 0 +
+e f“ 92, K,,—pKKH -— (92‘ K,_—pKK,_ dF(Z,|ZO)

— [K.. > K..- 1p; (K... — K..- )+ [K.. < K8118; (K..- — K...) (49)

 

 

 

=G(KO+,ZO)+e—WP(K0+,ZO)—e'rA’C(KO+,ZO)
_[K0+ > Ko—lpk(K0+ " K0—)+[K0+ < KO— 1P; (Ko— _ Km)

Here,

. A
G( . )-_- AKZ5’9KS: At +60%” 1" —8-"—Z,""1<{,’,aIF(Zl |Z0 ), (50a)

1‘ '.- A. _ _ A. _ _
p(.)___ f4 {Eh—Z" 9K: _ p1. [PE—[79— Z.‘ (21¢: _ pKKl_]}dF(Zl|ZO), (50b)

and

 

A, A, _ +
C(')= [H 6" Z.':"Kr: _p’ZK'i'i—é—Z" ”K.‘i—p1,K1-]}dF(Z.1Zo)- (509)

G( - ) is the PDV of the net cash flow for the so-called naive case. P( - ) is the

value of future disinvestment as the put option. The first bracket of function P is the net

79

.. -—-_-.

 

C"...

 

cash flow from the capital stock after disinvestment, K... The second bracket of function
P is the net cash flow from the capital stock before disinvestment, K 1-. Therefore, the
brace as a whole is the net gain from second period disinvestment. If the economic

indicator variable at the beginning of the second period, Z., is below the striking value,

y,‘ K ,_ , the firm will disinvest. Otherwise, the firm will not disinvest so that the net gain

is zero. Thus, function P is the expected net gain from second period disinvestment

 

conditional upon information available at the beginning of the first period, Z0. Similarly, h

if the economic indicator exceeds the striking price, y: K ,_ , the firm invests. Otherwise,

the firm does not invest. Function C( - ) is equal to the expected net gain from second

period investment conditional upon Z0, and the value of future investment as the call : '-
w

option.
A-3. Optimal Investment for First Period

The optimal decision for first period investment is to choose K,, to maximize V( - ).

mimosa )1 (51)

Equivalently, the optimal K0+ is determined from the derivatives of equation (49).

 

6V
°K,,Z 2 =0
‘1( 0 0)[ (3ij

or. by defining N(K0+,ZO) = 6G(K0+,Z0 )/aK,, ,

q°(Ko. ,ZO ) = N(1<0.,Z0 )+ e—WP'(KO.,ZO)— e-WC'(K0..Z0 )- p; = 0 (52a)
for investment, or

q°(Ko. .20): N(Ko.,Z0)+ e‘WP'(K0.,Zo )— e“”"C'(K0.,Zo)— p1} = 0 (52b)
for disinvestment.

Letting At = 1, equations N( - ), P’(- ), and C '(- ) can be expressed as follows:

 

N(-)= 4E 13"] , (53a)

80

 

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0'2
(90';
_ A'VEXPI:(1- 6{#Z — 2 ]_ 66] X (53b)
02 K0+ 9

and

c'(-)= e ”Z," (yr)“’(1—e-2"‘”) o. —e-"‘p;{1—q>['“(yiK0+/Z°)'5-(#z ”mm

0'2
+ A, expl:(1— 6(gz — 6:? )—66] x

{1_ ¢[ln(y,+1<.,./ZO)— {#2 - 5 — (9 — 1/2)afl]}[_20_]1_0.

 

 

 

02 K 0+
(53c)

When the depreciation is assumed to be zero as in ADEP (1996), the first terms in P'( - )
and C '( - ), which are functions of K0. rather than Z0 / Km, vanish. Then, all of three
functions become functions of Zn / K0,. By defining yo as Z0 / Km, and q(y(,) as N(yo) +
e" {P'(y(,)-C'(y(,)}, equations (52a) and (52b) can be rewritten as

q°(yo ) = {N(yo )- p}: }+ 6" {P'(yo )- C '(yo )i = 0, 0r (1(y0 ) = 19; (54a)
for investment, and

q°(yo)= {N (yo )- pk }+ e" {P'(yo )- C '(yo )i = 0. 0r q(yo ) = 19; (54b)
for disinvestment.

Figure A.1 shows the functions q(y), N(y), P'(y), and C '(y). In figure A.1, a
dashed line shows the function N(y), which is concave and monotonically increasing.
N(y) is the so-called naive case and the same as the derivative of the expected PDV of net
cash flow for the second period since the firm will not make any investment decision

after t = At. The function P'(y) appears as a convex function at the lower left comer. Its

81

 

 

y-intercept is close to but lower than p; ( = l), and it is monotonically decreasing and
approaches zero. The function C'(y) appears as a concave function in the lower right part
of the figure. It starts at the origin and remains close to zero until y approaches about 0.5.
Then, it increases as y increases. the function q(y) starts at the P'(y)’s y-intercept, and
moves horizontally to the right. Then, it moves along N(y), while P'(y) and CO») are

close to zero. Finally, it moves away from N(y) and crosses the purchase price line

( p; = 2) from below. A bold part of q(y) is relevant to optimal investment.

 

CHY) I‘

 

C' (y)

 

 

Figure A.1 q(y), N(y) , P' (y), and C’ (y) for Two Period Model

(6: 0.4, A”: 0.29., 6: 0, 7: 0.05, #2 = 0.02, 02 = #2, I); = 2, pl; : 1)

82

Because the firm anticipates high possibilities of future investment when the
economy is booming, i.e., Z, is high, so is y, C '02) has higher values when y is high. On
the other hand, when the economy is in slump, the firm contemplates disinvestment, so
that P’(y) is high when 20 is low. As the physical deprecation is assumed zero, C’(y) is
constant at zero for a low y, while P'(y) is constant at zero for a high y. When the
depreciation rate is non-zero, C '(y) is constant at a positive level rather than zero for low
y, while P'(y) is constant at a positive level lower than C '(y) for high y. Therefore, q(y)

shifts right when depreciation exists. In other words, optimal capital stock is lower when

there is depreciation than when there is no depreciation.

Figures 12 and 13 show the optimal rule for investment and disinvestment,
respectively. Optimal investment, given the purchase price of capital, p; , is the amount
that makes Z0 / K0. equals y; , if Z0 / K0. exceeds y; . As figure A.2 shows, the function
q moves apart from N0») before it reaches the purchase price line. The critical value for

q(y), y; , is higher than the critical value for N(y), yf , so that given 20, the firm should
invest less when it considers the possibility of future investment than when it does not.
The optimal disinvestment rule is that the firm should disinvest by the amount that makes
Z0 / K0. equal to yg , when ZO / K0. is below yg . Contrary to the optimal investment rule,

as figure A.3 shows, q(y) and N(y) cross the resale price at the same point. In other
words, the consideration of future disinvestment does not affect the optimal

disinvestment decision in this case.

83

 

 

I
‘m‘u' ‘

 

 

 

Figure A.2 Optimal Rule for Investment in Two Period Model

(yr z 0.497, and y; z 0.778)

 

 

 

Figure A.3 Optimal Rule for Disinvestment in Two Period Model

(y; z0.158. y; ~0.158)

84

 

APPENDIX B

APPROXIMATION OF G SATISFYING J(R,G) = 0

This appendix derives the approximation of G satisfying J(R,G) = 0.
J(R, G) a R h(G)— G'"”h(G" )= 0 (55)

Here,

 

 

 

0* satisfies J(R,G) = 0, given R.
B-l. Approximation by Order of Exponents
This section derives an approximation by choosing a greater term from two terms in
either the denominator or the numerator in equations g(x) and 1 — g(x). Since G > 1 and
¢N<O<l—6<1<¢W
0 < G"’" <1 < 0"” < G“’" , and O < G""’" < 0"”) <1 < GT".
Therefore, by assuming that G‘”" is much greater than G” , 0"“ is much greater than
0"“) , we have
Gw" —GH) z Cw" , Gm" —G"’" z -G"”" , and G4”) —G"’" z —G“"'.
Also, assuming that G "’-" z 0, and G"”" z 0,
G"’" —G"’"’ z Gq’" , G”) —G"’" z 6"”, and 6”" —G"”’ z —G"*”.
Then,

~l~0

l—0— -
= G "" z 0,
G w"

8(G)==v' 1. 1-g(G)~

 

85

 

G—l+()
—I ~
g<G >~ Gm

h(G)=———1——{1—-1—:2}, and h(G")=—1——{1—1—_—6}.

f(1-9) ¢~ f(1-<9) 4),,

Thus. equation (55) can be approximately written as

:G§1’\‘-l+0z 0, 1-g(G—l)z 1,

 

R (D,.—1+9— 0"” (pp—1+6
f(1-9) m f(1-9) 90,»

By rewriting equation (56), we have

Ga: z [[M][&)R:ll_o . (57)
¢P " 1 + 6 (PM

B-2. Approximation by Binomial Series

 

= 0. (56)

The Binomial series is

(1”). =1+¢x+¢(¢-1)x2 +¢(¢-1X¢-2)x3 +

2! 3! ... (5 8)

 

z1+¢x.
Then,
G"’" z1+¢,,(G—1), G"”z1+(1—6XG—l), G‘“ z1+¢,,(G—1),
G“"’" z1—¢,.(G—1). G‘W zl—(l—OXG—l), and G""’“ z1—¢,,,(G—1).
Therefore,

Gm. TC“ ”((01) ’(oN XG-l), Gm —Gl-0 z (901’ -1+9XG—1)’
0"" -G"’“ ”(l-B-(pNXG-I), 0"” -G“”"’ shop +¢~XG-1),
G""’" —G-'+” z(—go,, +1—6XG-1), and G“ —G"”" z(—1+6+¢NXG—1).

 

 

Thus,
.— 1—0— _
g(G)zfl——1:€ ~g(G"), and 1—g(G)z——-——fl z 1 —g(G '), and
(Pr- N ¢l’—¢N
— .— — 1—6— 6 _
f(1-6) (0N (Pr "‘ (0N (PP (PP '(DN
Asaresult,

86

 

R —G""= 0. (59)

By rewriting equation (59), we have

0" z RH) . (60)
B-3. Refinement of Approximation
Figure B] shows approximation of G* by equations (57) and (60), as well as the

approximation proposed by Abel and Eberly (1996) and exact G*. Abel and Eberly
suggest

I

G*~l+ 60'3 3(R 1)" 61
~ (1-6)(y+6) _. H ( )

 

 

 

 

 

Figure B] Approximation of G* (1)

( a: 0.4. 5: 0.06, 7: 0.051,;12 = 0.05, oz = 0.02)

87

Equation (57) seems a better fit, although 6* by equation (57) is not equal to

unity at R = 1. By adding an auxiliary term to equate G“ to 1 at R = 1, we have

I l
G*~ M £1; R W- (AV-1+6 (0" l_6+1 (62)
(pp-1+6 ¢N (D,.-1+9 (0N ’
l

_ 147)
or 0* z [MKQJM — 1)+1 . (63)
(pp -1+ 9 cm.

Figure B.2 shows 6* by equations (62) and (63) as well exact 6*. Equation (63) is a

 

better fit than equation (62). Equation (63) is equation (16) in the main text.

 

 

 

Figure B.2 Approximation of G* (2)

( 6: 0.4, a: 0.06, y: 0.051, #2 = 0.05. oz = 0.02)

88

 

APPENDIX C

FUNCTION q WITH AND WITHOUT FIXED COSTS AND
RANGE OF INACTION

This appendix discusses the coefficients of the function q, B,» and EN. In particular, this E‘
appendix shows 8,, < B; < 0 < Bi, < BN and y], < y‘ < y+ < y}, , where an asterisk (*)

indicates investment with fixed costs.

 

C-l. Coefficients of Function q, BN and Bp

The function q is written as follows:
q = Hy” + BNyq" + B,.y"’". (64)
H>0, BN>0, BP<O, 0< 6<1, (D,./<0, (0,.>1
Because of fixed costs, there are a triangular area below the resale price line and a

crescent area above the purchase price line for the function q’ withfixed costs. On the

other hand, the function q without fixed costs are tangential to those price lines.
Therefore, the fimction q' with fixed costs is below the function q without fixed costs for
small y, while the function q' is above the fimction q for large y. In other words,

q —- q. > 0 for small y, while q — q' < O for large y. The difference, q — q. , can be written

as follows:

q — q’ = (B. - 31);)“: + (13,. - 3;. )y:" . (65)
Since (0N < 0 and (0,. > 1, the first term of equation (65) dominates for small y, while the
second term of equation (65) dominates for large y. Therefore, BN — BR, > O to move the

local minimum of the function q below the resale price line, and Bp — B}. < 0 to move the

89

 

local maximum of the function q above the purchase price line. Thus, the coefficients of

the function q has the following relationship.

13,, < 3;, < 0 < 3;. < 13,, (66)
C-2. Range of Inaction
The function q first moves downward, then turns upward, and, moves downward again,

as y increases from zero. And, the boundary conditions for investment without fixed

costs is
q'(y)l= (1— 6)Hy"’ + <0~B~y"" “ + (pl-BM“ J= 0 at y“ and y*. (67)
Therefore, the derivative of q changes its sign from negative to positive to negative at the

critical values, y‘ and y’ . As a result, q’(y) is strictly positive for the range of inaction,

( y“ , y" ). The derivative of the function q' has the following relationship with the

function q.

q‘ '6) - q'ty) =60~ (3;. — 3., )y. " + q» (3;. — 3,, )y“"" > 0 fory > 0 (68)
At y" and y‘ , q. '02) > 0, since q'(y) = 0. Therefore, the positive slope of the function
q' is wider than that of the function q. In other words, by defining ' yg and y; at which
the function q' has the local minimum and the local maximum, respectively, we have
y; < y“ < y+ < y; since the slope of the function q' is strictly positive between y’ and

y+ . As a result, when investment has fixed costs, the local minimum for the function q

moves to the left and down, while the local maximum for the function q moves to the
right and upward. At the same time, at y}, and y}, , the function q. is negatively sloped.
Therefore, y}, < yg < y; < y;,. Thus, we have y}, < y' < y* < y;,. In other words, the
range of inaction for investment with fixed costs, ( y}, , y}, ), is wider than the inaction

range without fixed costs, ( y+ , y" ).

90

APPENDIX D
Serially Correlated Error

This appendix discusses a serially correlated error in the econometric models, and
Voung’s test of model selection is revised in accordance with a serial correlation

assumption. This appendix assumes that an error is an AR(1) process, i.e.,

u, = pu,_. + v, , where v, ~ i.i.d. N[O, a“2 ]. Discussion focuses on the OF M and OLS and
concludes that the GFM is still better than OLS.
D-l. Voung’s Test of Model Selection

With the serially correlated error, the likelihood function for the GFM becomes the

following.

L[u,|X, ,z,,n',vg]= n _1_¢[Lpfi] X H[<D( Xfll' - z,v). ]_ (pt/YEN (69)
. 0'

LI) 0' O- () 0-

 

Here, X, = {x,, z,,,,,,,.} and 77' = {77, v,-}. OLS, the censored and generalized Tobit models
and the RF M in the analysis have their own likelihood functions similar to equation (69).
This analysis uses estimates for u,, 77’, v,, and aunder the no-serial-correlation
assumption, since those estimates are consistent. This depends upon the fact that the
Tobit model under very general assumptions about u, is consistent (Robinson, 1982). The

estimator of p for this analysis is the following.

 

(70)

91

For t = 1 and terms after zero investment observations, a term in equation (69), u, — pu,_.,

is substituted with (1 - p)u,, since we do not have 11,-..

Table D.1 shows the log likelihood of five investment models. The GFM has the

highest likelihood among the five models regardless of the assumptions about serial

correlation. OLS is the second highest except the automobile industry under the no-

serial-correlation assumption. But, for the automobile industry under the no-serial-

correlation assumption, OLS is close to the G. Tobit model, which is the second best. We

conduct the Likelihood Ratio (LR) test. Its hypothesis is that the serial correlation

assumption and the no-serial-correlation assumption are equivalent, which implies that

there is no serial correlation in error. The GFM fails to reject the hypothesis at the ten

percent level in the computer and office equipment industry, at the five percent in the

airline industry, and at the one percent level in the automobile industry. OLS and the

RFM strongly reject the hypothesis in two industries, the automobile industry and the

Table D.1 Log Likelihood of Investment Models

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I OLS Tobit [ G. Tobit I RF M OF M
Computer and Office Equipment Industry
No Serial Correlation 62.397 45.693 59.658 52.502 71.366
Serial Correlation 64.458 47.044 60.270 54.663 71.508
LR Test 4.112 2.702 1.224 4.322 0.284
(0.081) (0.247) (0.611) (0.069) (0.915)
Automobile Industry
No Serial Correlation 73.004 43.434 74.923 57.450 89.875
Serial Correlation 78.912 45.918 76.616 62.795 92.338
LR Test 10.816 4.968 3.386 10.690 4.926
(0.000) (0.039) (0.148) (0.000) (0.040)
Airline Industry
No Serial Correlation 81.545 40.702 68.770 66.292 97.547
Serial Correlation 91.830 46.968 69.914 76.684 99.554
LR Test 20.590 12.532 2.288 20.784 4.014
(0.000) (0.000) (0.327) (0.000) (0.089)

 

 

 

 

 

 

Ho for LR test: the serial correlation assumption and the no serial correlation assumption

are equivalent.
p-value in parenthesis

92

 

 

airline industry. For the airline industry, the censored Tobit model also rejects the
hypothesis. The GFM and the G Tobit model show higher p-values than the other three
models.
Table D2 shows Voung’s test of model selection under the serial correlation
assumption. Voung’s test of model selection generally ranks the five models as follows:
GFM > OLS > G. Tobit > RFM > Tobit.

However, there is one exception, which is the G. Tobit model vs the RFM in the airline F“
industry. Even though this case has an opposite sign, the difference between the two E

competing models is statistically insignificant. In addition, OLS is as good as the GFM

 

t4. 4...

Table D2 Model Selection under Serially Correlated Error

1. Computer and Office Equipment Industry

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

F9 \ G, OLS Tobit G. Tobit RFM GFM
OLS 1.1887 0.7705 0.7095 0.2570
Tobit -2.062** 0.3553 1.1618 0.9699
G. Tobit -0.616 26.451 ** * 1.2263 0.4262
RFM -1.501 0.913 -0.654 0.6586
GFM 1.795* 3.207*** 2.223" 33.691***

2. Automobile Industry

F0 \ G, OLS Tobit G. Tobit RFM GFM
OLS 1.2654 0.5495 0.9537 0.6463
Tobit -3.885*** 0.6726 1.1400 1.8953
G. Tobit -0.410 61.396*** 0.7479 0.4882
RFM ~2.186** 2.094" -2.117** 0.7542
GFM 2.212“ 4.466*** 2.980*"‘* 59.085***

3. Airline Industry

F9 \ G, OLS Tobit G. Tobit RFM GFM
OLS 0.8462 0.5286 0.6315 0.3999
Tobit -5.594* ** 0.2626 0.8686 1.2781
G. Tobit -3.458*** 45.893*** 0.8517 0.6338
RFM -2.186** 3.658*** 0.841 0.6545
GFM 1.401 5.336*** 4.271*** 45.741***

 

 

 

 

 

 

 

 

 

Lower lefi: 2><LR(-) or (NT )‘V 2 LR(-)/ a}; a positive value favors F 9 over G,

Upper right: 6132
*: reject H0 at 10 %

**z reject H0 at 5 %

***: reject Ho at 1 %

in the airline industry. Although the GFM has a higher likelihood than OLS, the

difference between the GFM and OLS is statistically insignificant.

D-2. Estimated p and Dutbin-Watson Test of OLS Residuals

Table D.3 shows the estimated p which are used for the maximum likelihood estimation

under the serial correlation assumption. The GF M and the G. Tobit model show lower

estimates, while OLS, the Tobit model and the RFM show higher estimates. The GFM

ranks its estimated p as: the automobile industry > the airline industry > the computer and

office equipment industry. And, OLS ranks the estimated p as: the airline industry > the

automobile industry > the computer industry. These results are consistent with the test of

serial correlation in the main test.

Table D.4 shows the Durbin-Watson (D-W)statistic for the OLS residuals. The

D-W statistic is based upon the OLS residuals. The computer and office equipment

industry shows the highest statistic, while the airline industry shows the lowest statistic.

All of the three industries reject the no-serial-correlation hypothesis at five percent, since

their D-W statistic is lower the corresponding critical value, d), 5 9,. The results are also

compatible with the test of serial correlation.

Table D.3 Estimated Coefficient of AR(1) Process, ,0

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Industry OLS Tobit G. Tobit RFM GFM
Computer 0.2826 0.2915 0.1822 0.2854 0.0781
Automobile 0.5086 0.4918 0.2776 0.5042 0.3129
Airline 0.5563 0.5813 0.2255 0.5693 0.2514
Table D.4 Durbin-Watson Statistic for OLS Residuals
IndUStry D'W StatiStic kl, n d,“ 5% d“. 5%
Computer 1.3527 6, 60 1.372 1.808
Automobile 0.9778 6, 57 1.334 1.814
Airline 0.8491 8, 76 1.399 1.867

 

 

 

 

 

94

 

 

D-3. Discussion

Even though this appendix incorporates serial correlation in error into the analysis, the
GFM is still likely to be the best among the five models. The GFM residuals seem to be
serially uncorrelated. On the other hand, OLS residuals seem to be serially correlated. In
addition, OLS with serial correlation could be as good as the GF M when we look at the
airline industry. However, there is one reason to conclude that OLS is biased and the
GFM is better than OLS. Figure D] is a schematic diagram for comparison of the GFM
with OLS. In figure D. l , solid lines represent the GFM, while a broken line represents
OLS. As figure D.1 shows, OLS overestimates investment, while it underestimates
disinvestment. At the same time, Ln(Z, / K,) is assumed to be an arithmetic Brownian
motion, or a unit root process, so that, when it is high, it remains high for a while. Since
the OF M is assumed to be consistent, observations are located around the GFM, and each
GFM residual is either negative or positive regardless of Ln(Z, / K,). On the other hand,
when an OLS residualiis negative for high Ln(Z, / K,), the next OLS residual is likely to
be negative because Ln(Z, / K,) likely remains high due to the unit root process. Thus, the
OLS residuals are likely to remain negative as long as Ln(Z, / K.) is high. Therefore, the
OLS residuals are serially correlated with positive p, while the GF M residuals are serially

uncorrelated.

 

 

’/
019/
/’ 0
LnG‘ ——————————— 7”—
’ I
o ——————— r—o~éo—| GFM
/
LnG' —————— l/ :
I( I
/'I I
/ I l
(D) I (O) I (I)
Lnyér LnYi-r Ln (Zt/Kt-)

Figure D.1 GFM vs OLS

95

 

APPENDIX E

TWO STEP ESTIMATION FOR
LIMITED DEPENDENT VARIABLE MODEL

In order to conduct the test of serial correlation, we estimate the limited dependent
variable model by a two step estimation (TSE). This section shows its estimates. For
comparison, the GFM estimates are reported in the same functional form. Both estimates
seem to be close to each other.

For non-limited observations such as investment observations, k,, their expected

value, E [k, |x, , z, ,k, > 0], is written as the following.

E[k,|x, ,z, ,k, > 0] = z,v; + x,77 — z,.,,m.v_, + Elu, > —(z,v; + x,77 — z,.,,,,,.v., )J
¢[(le; + x!" _ ZIJchf) 01
(Dl(z,v; + x,77 — zhmv, )/0'J

= z,v; + x,q — z,',,,,cv, + ovi[(z,v; + x,77 e z,.,,,,,.v, )/0'] (71)

 

+
: Zlvg +x177-Zumcvf +0

Here, (15 and (D are the p.d.f. and the c.d.f. of the standard normal distribution,
respectively, and [1(-)(E ¢(-)/<D(o)) is the inverse Mills ratio. Disinvestment observations
have a similar expected value. The two step estimation used in the analysis first
estimates 11(-), and, then estimates coefficients in equation (71).

The first step of the two step estimation is a probit-type discrete regression model
using all observations, and estimates the inverse Mills ratio, l(-). The likelihood function
for the model is the following:

L = 1,1(1 - <Dl(- M + zwv/ )/0D
x [J(CDK— x,7] + z,.,,,,,.v_, )/a']— (D[(-— x,77 + z,.,,.,,.vq, — 2, v, )/0'D ' (72)
x §(¢[(-xm+ - ram/01)

96

The second step is OLS of the dependent variable, k,, on the explanatory

variables, x,, z,,,,,,,, and the estimated inverse Mills ration, 2: , using only investment
observations and disinvestment observations. Although TSE’s standard errors are
reported in parentheses, they are invalid since the TSE is a regression on a generated
regressor. Brackets in the following equations are an index variable which gives one if
observation is disinvestment, or zero otherwise, i.e., [disinvestment] = 1 if an observation
is disinvestment, 0 if an observation is investment. The TSE estimates and the GFM

estimates are the following.

(a) Computer and Office Equipment Industry

TSE 12, = 0.5867 Re, — 0.3654 K, — 0.2426 (wL),
(0.1438) (0.0653) (0.0824)

— 0.3874 d1 — 0.4226 d2 — 0.3272 d3 + 0.0183 7+ 0.0382 2
(0.3056) (0.3093) (0.3407) (0.0063) (0.0410)
— [disinvestment](0.2318 d1 + 0.2318 d2 + 0.2318 d3 + 0.0382 y)
(0.1742) (0.1742) (0.1742) (0.0213)

GFM 12, = 0.5586 Re, — 0.3508 K, — 0.2265 (wL),
(0.1290) (0.0582) (0.0740)
— 0.3661 d1 — 0.4108 d2 — 0.3068 d3 + 0.0178 7
(0.2797) (0.2837) (0.3121) (0.0058)
— [disinvestment](0.0758 d1 + 0.0758 d2 + 0.0758 d3 + 0.0260 7)

(0.2603) (0.2658) (0.2950) (0.0135)
6:0.0707 (0.0066)

(b) Automobile Industry

TSE 12, = 0.1689 Re, — 0.0998 K, — 0.0525 (wL),
(0.1762) (0.0260) (0.2215)

— 0.1801 d1 — 0.2367 d2 — 0.2042 d3 + 0.0079 y— 0.0454 ,1
(0.4547) (0.5085) (0.5218) (0.0044) (0.0229)

- [disinvestment](— 0.0999 d1 — 0.0999 d2 - 0.0999 d3 — 0.0081 y)

(0.1446) (0.1446) (0.1446) (0.0085)
GFM 12,: 0.1627 Re, — 0.1097 K, - 0.0245 (wL),

(0.1659) (0.0241) (0.2083)

— 0.3198 d1 -— 0.3866 d2 — 0.3607 d3 + 0.0074 y
(0.4232) (0.4738) (0.4859) (0.0042)

— [disinvestment](0.1775 d1 + 0.1775 d2 + 0.1775 d3 — 0.0089 7)

(0.4395) (0.4902) (0.5023) (0.0091)
6' = 0.0458 (0.0044)

97

 

(c) Airline Industry

TSE 12,: 0.2788 Re, — 0.0952 K, — 0.1964 (wL), + 0.1906 d] + 0.1399 d2
(0.1422) (0.0467) (0.1133) (0.3578) (0.3483)

+ 0.1465 d3 + 0.2218 d4 — 0.0086 dereg + 0.0026 7— 0.0283 ,1
(0.3647) (0.3161) (0.1251) (0.0044) (0.0197)
— [disinvestment]( - 0.0318 d1 - 0.0318 d2 — 0.0318 d3
(0.1203) (0.1203) (0.1203)
— 0.0318 d4 — 0.0318 dereg + 0.0132 ;)
(0.1203) (0.1203) (0.0127)

GFM 1;, = 0.2450 Re, — 0.0963 K, — 0.1747 (wL), + 0.3046 d1 + 0.2448 d2
(0.1303) (0.0434) (0.1043) (0.3243) (0.3164)
+ 0.2541 d3 + 0.3167 d4 + 0.4427 dereg + 0.0031 y
(0.3317) (0.2873) (34.672) (0.0041)
— [disinvestment](0.1333 d1 + 0.1333 d2 + 0.1333 d3
(0.3348) (0.3268) (0.3418)
+ 0.1333 d4 + 0.1333 dereg + 0.0159 7)
(0.2973) (33.841) (0.0107)
6 = 0.0636 (0.0053)

For the three industries, both the TSE estimates and the GFM estimates seem to be close

to each other.

98

 

BIBLIOGRAPHY

99

BIBLIOGRAPHY

Abel, Andrew B., “Optimal Investment under Uncertainty.” American Economic Review,
March 1983 73(1), pp. 228-33. ‘

, and Blanchard, Oliver J ., “The Present Value of Profits and Cyclical Movement
in Investment.” Econometrica, March 1986, 54(2), pp. 249-73.

, Dixit, Avinash K., Eberly, Janice C. and Pindyck, Robert 8., “Options, the Value
of Capital, and Investment.” Quarterly Journal of Economics, August 1996,
111(3), pp. 753-77.

, and Eberly, Janice C., “A Unified Model of Investment under Uncertainty.”
American Economic Review, December 1994, 84(5), pp. 1369-84.

, and , “Investment and q with Fixed Costs: An Empirical Analysis.”
mimeo, January 1996.

, and , “Optimal Investment with Costly Reversibility.” Review of Economic
Studies, October 1996, 63(4), pp. 581-593.

, and , “The Mix and Scale of Factors with Irreversibility and Fixed Costs of
Investment.” Carnegie-Rochester Public Policy Conference, April 1997.

Amemiya, Takeshi, Advanced Econometrics, Cambridge: Harvard University Press,
1985.

Arrow, Kenneth J ., “Optimal Capital Policy with Irreversible Investment.” In James N.
Wolfe ed., Value, Capital and Growth. Papers in Honour of Sir John Hicks.
Edinburgh: Edinburgh University Press, 1968, pp. 1-19.

Black, Fischer, and Scholes, Myron, “The Pricing of Options and Corporate Liabilities.”
Journal of Political Economy, May-June 1973, 81(3), pp. 637-59.

Bemanke, Ben S., “Irrevesibility, Uncertainty, and Cyclical Investment.” Quarterly
Journal of Economics, February 1983, 98(1), pp. 85-106.

Bertola, Guiseppe and Caballero, Ricardo J ., “Irreversibility and Aggregate Investment.”
Review of Economic Studies, April 1994, 61 (207), pp. 223-46.

Brealey, Richard A. and Myers, Stewart C., Principles of Corporate Finance. 4th ed.
New York: McGraw-Hill, 1992.

Brennan, M.J., “Capital Asset Pricing Model.” In John Eatwell, Murray Milgate and
Peter Newman ed.. The New Palgrave: A Dictionary of Economics. New York:
W.W. Norton, 1987, pp. 91-102.

Caballero, Ricardo J ., “On the Sign of the Investment-Uncertainty Relationship.”
American Economic Review, March 1991, 81(1), pp. 279-88.

100

Copeland, Thomas E. and Weston, J. Fred, “Asset Pricing.” In John Eatwell, Murray
Milgate and Peter Newman ed.. The New Palgrave: A Dictionary of Economics.
New York: W.W. Norton, 1987, pp. 81-85.

Craine, Roger, “Risky Business: The Allocation of Capital.” Journal of Monetary
Economics, March 1989, 23(2), pp. 201-18.

Dagenais, Marcel G., “A Threshold Regression Model.” Econometrica, April 1969,
37(2), pp. 193-203.
, “Application of a Threshold Regression Model to Household Purchases of
Automobiles.” Review of Economics and Statistics, August 1975, 5 7(3), pp. 275-
85.

Dixit, Avinash K. and Pindyck, Robert 8., Investment Under Uncertainty. Princeton:
Princeton University Press, 1994.

Fishe, Raymond RH, and Lahari, Kajal, “On the Estimation of Inflationary Expectations
from Qualitative Responses.” Journal of Econometrics, May 1981, 16(1), pp. 89-
102.

Fisher, Jonas D.M., and Homstein, Andreas, “(S,s) Inventory Policies in General
Equilibrium.” Federal Reserve Bank of Chicago Working Paper, December 1996.

Gould, John P., “Adjustment Costs in the Theory of Investment of the Firm.” Review of
Economic Studies, January 1968, 35(101), pp. 47-55.

Guiso, Luigi and Parigi, Giuseppe, “Investment and Demand Uncertainty.” Bank of
Italy Working Paper, No. 289, November 1996. '

Hamermesh, Daniel S, and Pfann, Gerard A., “Adjustment Costs in Factor Demand.”
Journal of Economic Literature, September 1996, 34(3), pp. 1264-92.

Hartman, Richard, “The Effect of Price and Cost Uncertainty on Investment.” Journal of
Economic Theory, October 1972, 5(2), pp. 258-66.

Hayashi, F umio, “Tobin’s Marginal q and Average q: A Neoclassical Interpretation.”
Econometrica, January 1982, 50(1), pp. 213-24.

Honore, Bo E., “Orthogonality Conditions for Tobit Models with Fixed Effects and
Lagged Dependent Variables.” Journal of Econometrics, September 1993, 5 9(1-
2), pp. 35-61.

Huberman, Gur, “Arbitrage Pricing Theory.” In John Eatwell, Murray Milgate and Peter
Newman ed.. The New Palgrave: A Dictionary of Economics. New York: W.W.
Norton, 1987, pp. 72-80.

101

 

Ingersoll, Jonathan E. Jr., “Option Pricing Theory.” In John Eatwell, Murray Milgate
and Peter Newman ed.. The New Palgrave: A Dictionary of Economics. New
York: W.W. Norton, 1987, pp. 199-212.

J akubson, George, “The Sensitivity of Labor-Supply Parameter Estimates to Unobserved
Individual Effects: F ixed- and Random-Effects Estimates in a Nonlinear Model
Using Panel Data.” Journal of Labor Economics, July 1988, 6(3), pp. 302-329.

Jorgenson, Dale W., “Capital Theory and Investment Behavior.” American Economic
Review, May 1963 (Papers and Proceedings), 53(2), pp. 247-59.

Leahy, John V. and Whited, Toni M., “The Effect of Uncertainty on Investment: Some
Stylized Facts.” Journal of Money, Credit, and Banking, February 1996, 28(1),
pp. 64-83.

Lucas, Robert E. Jr., “Optimal Investment with Rational Expectations.” In Robert E.
Lucas Jr., and Thomas J. Sargent, ed., Rational Expectations and Economic
Practice, Vol 1. Minneapolis: University of Minneapolis Press, 1981, pp. 55-66.

, “Adjustment Costs and the Theory of Supply.” Journal of Political Economy,
August 1967, 75(4), pp. 321-34.
, “Optimal Investment Policy and the Flexible Accelerator.” International
Economic Review, February 1967, 8(1), pp. 78-85.
and Prescott, Edward, “Investment under Uncertainty.” Econometrica, September
1971, 39(5), pp. 659-81.

——_—.

Maddala, G. S., Limited-Dependent and Qualitative Variables in Econometrics. New
York: Cambridge University Press, 1983.

Malliaris, A. G. and Brook, W. A., Stochastic Methods in Economics and Finance. New
York: North-Holland, 1982.

McDonald, Robert and Siegel, Daniel, “The Value of Waiting to Invest.” Quarterly
Journal of Economics, November 1986, 101(4), pp. 707-27.

Merton, Robert C., “Applications of Option-Pricing Theory: Twenty-Five Years Later.”
American Economic Review, June 1998, 88(3), pp. 323-49.

, Continuous-Time Finance. Cambridge: Blackwell, 1990.

Mussa, Michael, “External and Internal Adjustment Costs and the Theory of Aggregate
and Firm Investment.” Economica, May 1977, 44(174), pp. 163-78.

Robinson, Peter M., “On the Asymptotic Properties of Estimators of Models Containing
Limited Dependent Variables.” Econometrica, January 1982, 50(1), pp. 27-41.

Rosett, Richard N., “A Statistical Model of Friction in Economics.” Econometrica,
April 1959, 27(2), pp. 263-67.

102

 

and Nelson, Forrest D., “Estimation of the Two-Limit Probit Regression Model.”
Econometrica, January 1975, 43(1), pp. 141-46.

Rothschild, Michael, “On the Cost of Adjustment.” Quarterly Journal of Economics,
November 1971, 85(4), pp. 605-22.

Scholes, Myron S., “Derivatives in a Dynamic Environment.” American Economic
Review, June 1998, 88(3), pp. 350-70.

Tobin, James, “Estimation of Relationships for Limited Dependent Variable.”
Econometrica, January 195 8, 26(1), pp. 24-36.
“A general Equilibrium Approach to Monetary Theory.” Journal of Money,
Credit, and Banking, February 1969, 1(1), pp. 15-29.

Treadway, A.B., “On Rational Entrepreneurial Behavior and the Demand for
Investment.” Review of Economic Studies, April 1969, 3 6(106), pp. 227-39.

Varian, Hal R., Microeconomic Analysis. 3rd ed. New York: W.W. Norton, 1992.

Voung, Quang H., “Likelihood Ratio Test for Model Selection and Non-nested
Hypothesis.” Econometrica, March 1989, 57(2), pp. 307-333.

Wooldridge, Jeffrey M., Econometric Analysis of Cross Section and Panel Data. 1995.

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