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Library of Congress Cataloging-in-Publication Data:

Panik, Michael J.
Growth curve modeling : theory and applications / Michael J. Panik.
pages cm
Includes bibliographical references and index.
ISBN 978-1-118-76404-6 (cloth)
1. Mathematical statistics. 2. Time series analysis. 3. Regression analysis.
4. Multivariate analysis. I. Title.
QA276.P2243 2013
519.5–dc23

2013026535

In memory of Michael Christopher Duffy

CONTENTS

PREFACE

1 MATHEMATICAL PRELIMINARIES

1.1 ARITHMETIC PROGRESSION

1.2 GEOMETRIC PROGRESSION

1.3 THE BINOMIAL FORMULA

1.4 THE CALCULUS OF FINITE DIFFERENCES

1.5 THE NUMBER E

1.6 THE NATURAL LOGARITHM

1.7 THE EXPONENTIAL FUNCTION

1.8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS: ANOTHER LOOK

1.9 CHANGE OF BASE OF A LOGARITHM

1.10 THE ARITHMETIC (NATURAL) SCALE VERSUS THE LOGARITHMIC SCALE

1.11 COMPOUND INTEREST ARITHMETIC

2 FUNDAMENTALS OF GROWTH

2.1 TIME SERIES DATA

2.2 RELATIVE AND AVERAGE RATES OF CHANGE

2.3 ANNUAL RATES OF CHANGE

2.4 DISCRETE VERSUS CONTINUOUS GROWTH

2.5 THE GROWTH OF A VARIABLE EXPRESSED IN TERMS OF THE GROWTH OF ITS INDIVIDUAL ARGUMENTS

2.6 GROWTH RATE VARIABILITY

2.7 GROWTH IN A MIXTURE OF VARIABLES

3 PARAMETRIC GROWTH CURVE MODELING

3.1 INTRODUCTION

3.2 THE LINEAR GROWTH MODEL

3.3 THE LOGARITHMIC RECIPROCAL MODEL

3.4 THE LOGISTIC MODEL

3.5 THE GOMPERTZ MODEL

3.6 THE WEIBULL MODEL

3.7 THE NEGATIVE EXPONENTIAL MODEL

3.8 THE VON BERTALANFFY MODEL

3.9 THE LOG–LOGISTIC MODEL

3.10 THE BRODY GROWTH MODEL

3.11 THE JANOSCHEK GROWTH MODEL

3.12 THE LUNDQVIST–KORF GROWTH MODEL

3.13 THE HOSSFELD GROWTH MODEL

3.14 THE STANNARD GROWTH MODEL

3.15 THE SCHNUTE GROWTH MODEL

3.16 THE MORGAN–Mercer–FLODIN (M–M–F) GROWTH MODEL

3.17 THE MCDILL–AMATEIS GROWTH MODEL

3.18 AN ASSORTMENT OF ADDITIONAL GROWTH MODELS

APPENDIX 3.A THE LOGISTIC MODEL DERIVED

APPENDIX 3.B THE GOMPERTZ MODEL DERIVED

APPENDIX 3.C THE NEGATIVE EXPONENTIAL MODEL DERIVED

APPENDIX 3.D THE VON BERTALANFFY AND RICHARDS MODELS DERIVED

APPENDIX 3.E THE SCHNUTE MODEL DERIVED

APPENDIX 3.F THE MCDILL–AMATEIS MODEL DERIVED

APPENDIX 3.G THE SLOBODA MODEL DERIVED

APPENDIX 3.H A GENERALIZED MICHAELIS–MENTEN GROWTH EQUATION

4 ESTIMATION OF TREND

4.1 LINEAR TREND EQUATION

4.2 ORDINARY LEAST SQUARES (OLS) ESTIMATION

4.3 MAXIMUM LIKELIHOOD (ML) ESTIMATION

4.4 THE SAS SYSTEM

4.5 CHANGING THE UNIT OF TIME

4.6 AUTOCORRELATED ERRORS

4.7 POLYNOMIAL MODELS IN T

4.8 ISSUES INVOLVING TRENDED DATA

APPENDIX 4.A OLS ESTIMATED AND RELATED GROWTH RATES

5 DYNAMIC SITE EQUATIONS OBTAINED FROM GROWTH MODELS

5.1 INTRODUCTION

5.2 BASE-AGE-SPECIFIC (BAS) MODELS

5.3 ALGEBRAIC DIFFERENCE APPROACH (ADA) MODELS

5.4 GENERALIZED ALGEBRAIC DIFFERENCE APPROACH (GADA) MODELS

5.5 A SITE EQUATION GENERATING FUNCTION

5.6 THE GROUNDED GADA (G-GADA) MODEL

APPENDIX 5.A GLOSSARY OF SELECTED FORESTRY TERMS

6 NONLINEAR REGRESSION

6.1 INTRINSIC LINEARITY/NONLINEARITY

6.2 ESTIMATION OF INTRINSICALLY NONLINEAR REGRESSION MODELS

APPENDIX 6.A GAUSS–NEWTON ITERATION SCHEME: THE SINGLE PARAMETER CASE

APPENDIX 6.B GAUSS–NEWTON ITERATION SCHEME: THE R PARAMETER CASE

APPENDIX 6.C THE NEWTON–RAPHSON AND SCORING METHODS

APPENDIX 6.D THE LEVENBERG–MARQUARDT MODIFICATION/COMPROMISE

APPENDIX 6.E SELECTION OF INITIAL VALUES

7 YIELD–DENSITY CURVES

7.1 INTRODUCTION

7.2 STRUCTURING YIELD–DENSITY EQUATIONS

7.3 RECIPROCAL YIELD–DENSITY EQUATIONS

7.4 WEIGHT OF A PLANT PART AND PLANT DENSITY

7.5 THE EXPOLINEAR GROWTH EQUATION

7.6 THE BETA GROWTH FUNCTION

7.7 ASYMMETRIC GROWTH EQUATIONS (FOR PLANT PARTS)

APPENDIX 7.A DERIVATION OF THE SHINOZAKI AND KIRA YIELD–DENSITY CURVE

APPENDIX 7.B DERIVATION OF THE FARAZDAGHI AND HARRIS YIELD–DENSITY CURVE

APPENDIX 7.C DERIVATION OF THE BLEASDALE AND NELDER YIELD–DENSITY CURVE

APPENDIX 7.D DERIVATION OF THE EXPOLINEAR GROWTH CURVE

APPENDIX 7.E DERIVATION OF THE BETA GROWTH FUNCTION

APPENDIX 7.F DERIVATION OF ASYMMETRIC GROWTH EQUATIONS

APPENDIX 7.G CHANTER GROWTH FUNCTION

8 NONLINEAR MIXED–EFFECTS MODELS FOR REPEATED MEASUREMENTS DATA

8.1 SOME BASIC TERMINOLOGY CONCERNING EXPERIMENTAL DESIGN

8.2 MODEL SPECIFICATION

8.3 SOME SPECIAL CASES OF THE HIERARCHICAL GLOBAL MODEL

8.4 THE SAS/STAT NLMIXED PROCEDURE FOR FITTING NONLINEAR MIXED-EFFECTS MODEL

9 MODELING THE SIZE AND GROWTH RATE DISTRIBUTIONS OF FIRMS

9.1 INTRODUCTION

9.2 MEASURING FIRM SIZE AND GROWTH

9.3 MODELING THE SIZE DISTRIBUTION OF FIRMS

9.4 GIBRAT’S LAW (GL)

9.5 RATIONALIZING THE PARETO FIRM SIZE DISTRIBUTION

9.6 MODELING THE GROWTH RATE DISTRIBUTION OF FIRMS

9.7 BASIC EMPIRICS OF GIBRAT’S LAW (GL)

9.8 CONCLUSION

APPENDIX 9.A KERNEL DENSITY ESTIMATION

APPENDIX 9.B THE LOG-NORMAL AND GIBRAT DISTRIBUTIONS

APPENDIX 9.C THE THEORY OF PROPORTIONATE EFFECT

APPENDIX 9.D CLASSICAL LAPLACE DISTRIBUTION

APPENDIX 9.E POWER-LAW BEHAVIOR

APPENDIX 9.F THE YULE DISTRIBUTION

APPENDIX 9.G OVERCOMING SAMPLE SELECTION BIAS

10 FUNDAMENTALS OF POPULATION DYNAMICS

10.1 THE CONCEPT OF A POPULATION

10.2 THE CONCEPT OF POPULATION GROWTH

10.3 MODELING POPULATION GROWTH

10.4 EXPONENTIAL (DENSITY-INDEPENDENT) POPULATION GROWTH

10.5 DENSITY-DEPENDENT POPULATION GROWTH

10.6 BEVERTON–HOLT MODEL

10.7 RICKER MODEL

10.8 HASSELL MODEL

10.9 GENERALIZED BEVERTON–HOLT (B–H) MODEL

10.10 GENERALIZED RICKER MODEL

APPENDIX 10.A A GLOSSARY OF SELECTED POPULATION DEMOGRAPHY/ECOLOGY TERMS

APPENDIX 10.B EQUILIBRIUM AND STABILITY ANALYSIS

APPENDIX 10.C DISCRETIZATION OF THE CONTINUOUS-TIME LOGISTIC GROWTH EQUATION

APPENDIX 10.D DERIVATION OF THE B–H S–R RELATIONSHIP

APPENDIX 10.E DERIVATION OF THE RICKER S–R RELATIONSHIP

APPENDIX A

REFERENCES

INDEX

PREFACE

The concept of growth is all-pervasive. Indeed, issues concerning national economic growth, human population growth, agricultural/forest growth, the growth of firms as well as of various insect, bird, and fish species, and so on, routinely capture our attention. But how is such growth modeled and measured?

The objective of this book is to convey to those who attempt to monitor the change in some variable over time that there is no “one-size-fits-all” approach to growth measurement; a growth model useful for studying an agricultural crop will most assuredly not be appropriate for fishery management. And if, for instance, one is interested in calculating a growth rate for some time series data set, a decision has to be made as to whether or not one needs to determine a relative rate of growth, an average annual growth rate, an ordinary least squares growth rate, a geometric mean growth rate, among others. Moreover, the choice of a growth rate is subject to the idiosyncrasies of the data set itself, for example, we need to ask if the data series is trended or if it is stationary and if it is presented on an annual, a quarterly, or monthly basis. But this is not the whole story—we also need to ask if the appropriate growth curve should be linear, sigmoidal (S-shaped), with an upper asymptote, or, say, increases to a maximum and then decreases thereafter.

The aforementioned issues concerning the selection of a growth modeling methodology are of profound importance to those looking to develop sound growth measurement techniques. This book is an attempt to point them in the appropriate direction. It will appeal to students and researchers in a broad spectrum of activities (including business, government, economics, planning, medical research, resource management, among others) and presumes that the reader has had an elementary calculus course along with some exposure to basic statistical analysis. While derivations of virtually all of the major growth curves/models have been provided, they have been placed into end-of-chapter appendices so as not to interrupt the general flow of the material. Some important features of this book are: (i) in addition to detailed discussions of growth modeling/theory, the requisite mathematical and statistical apparatus needed to study the same is provided; (ii) SAS code (SAE/ETS 9.1, 2004) is given so that the reader can estimate their own specialized growth rates and curves; and (iii) an assortment of important applications are supplied.

Looking to specifics:

Although this project was initiated while the author was teaching at the University of Hartford, West Hartford, CT, the manuscript was completed over a number of years during which the author was Visiting Professor of Mathematics at Trinity College, Hartford, CT. A sincere thank you goes to my colleague Farhad Rassekh at the University of Hartford for all of our illuminating discussions concerning growth issues and methodology. His support and encouragement is greatly appreciated. I also wish to thank Paula Russo of Trinity College for allowing me to avail myself of the resources of the Mathematics Department.

A special thank you goes to Alice Schoenrock for all of her excellent work during the various phases of the preparation of the manuscript. Her timely response to a whole list of challenges is most admirable.

An additional note of appreciation goes to Susanne Steitz-Filler, Editor, Mathematics and Statistics, at John Wiley & Sons, for her professionalism, vision, and effort expended in the review and approval processes.

1

MATHEMATICAL PRELIMINARIES

1.1 ARITHMETIC PROGRESSION

We may define an arithmetic progression as a set of numbers in which each one after the first is obtained from the preceding one by adding a fixed number called the common difference. Suppose we denote the common difference of an arithmetic progression by d, the first term by a₁, …, and the nth term by a_n. Then the terms up to and including the nth term can be written as

(1.1)

If S_n denotes the sum of the first n terms of an arithmetic progression, then

(1.2)

If the n terms on the right-hand side of Equation 1.2 are written in reverse order, then S_n can also be expressed as

(1.3)

Upon adding Equations 1.2 and 1.3, we obtain

or

(1.4)

EXAMPLE 1.1 Given the arithmetic progression –3, 0, 3, …, determine the 50th term and the sum of the first 100 terms. For a₁ = –3, the second term (0) minus the first term is 0 – (–3) = 3 = d, the common difference. Then, from Equation 1.1,

and, from Equation 1.4,

1.2 GEOMETRIC PROGRESSION

A geometric progression is any set of numbers having a common ratio; that is, the quotient of any term (except the first) and the immediately preceding term is the same. Suppose we represent the common ratio of a geometric progression by r, the first term by a₁(≠0), ∆, and the nth term by a_n. Then the terms up to and including the nth term are

(1.5)

(Note that, as required,

If the sum of the first n terms of a geometric progression is denoted as S_n, then

(1.6)

Using Equation 1.6, let us form

(1.7)

so that, upon subtracting Equation 1.7 from Equation 1.6, we obtain

or

(1.8)

EXAMPLE 1.2 Given the geometric progression 1/2, 3/4, 9/8, …, determine the sixth term and the sum of the first nine terms. For a₁ = 1/2, the second term (3/4) divided by the first term (1/2) is (3/4)/(1/2) = 3/2 = r, the common ratio. Then, from Equation 1.5,

and, from Equation 1.8,

Suppose we have a geometric progression with infinitely many terms. The sum of the terms of this type of geometric progression, in which the value of n can increase without bound, is called a geometric series and has the form

(1.9)

If we again designate the sum of the first n terms in Equation 1.9 as S_n (here S_n is called a finite partial sum of the first n terms) or Equation 1.6, then, via Equation 1.8,

(1.10)

If |r| < 1, then the second term in the difference on the right-hand side of Equation 1.10 decreases to zero as n increases indefinitely (rⁿ → 0 as n → ∞). Hence,

(1.11)

Thus, the geometric series S is said to converge to the value a₁/(1 – r). If |r| > 1, the finite partial sums S_n do not approach any limiting value—the geometric series S does not converge; it is said to diverge since |rⁿ| → ∞ as n → ∞.

EXAMPLE 1.3 Given the geometric progression

does the geometric series

converge? If so, to what value? Given r = 1/3, the nth finite partial sum is

and, via Equation 1.10,

Then

1.3 THE BINOMIAL FORMULA

Suppose we are interested in finding (a + b)ⁿ, where n is a positive integer. According to the binomial formula,

(1.12)

with the coefficients of the terms on the right-hand side of Equation 1.12 termed binomial coefficients corresponding to the exponent n. For instance, from Equation 1.12,

Note that, in general:

A glance back at Equation 1.12 reveals that the (r + 1)st term in the binomial expansion of (a + b)ⁿ is

(1.13)

That is, for

If, as in the preceding text, n = 5, then the preceding three binomial expansion terms are

Given Equation 1.13, we can now write the general binomial expansion formula as

(1.14)

1.4 THE CALCULUS OF FINITE DIFFERENCES

Suppose that the real-valued function y = f(x) is defined on an interval containing x and Δx (i.e., x has been increased by an amount Δx). Since the difference interval Δx is generally a constant, we may simply denote this constant as h. Then the difference operator Δ applied to f(x) is defined as

(1.15)

Furthermore, while h may be any constant value, it is usually the case that h = 1. Hence, in what follows, the interval of differencing in x is unity. Thus, Equation 1.15 becomes

(1.15.1)

Given Equation 1.15.1, it is readily verified that:

(1.16)

Clearly

For real-valued functions f(x) and g(x) both defined over an interval containing x and x + 1,

(1.17)

Here

(Note that if c₁ and c₂ are arbitrary constants, then Δ[c₁f(x) ± c₂g(x)] = c₁Δf(x) ± c₂Δg(x).)

(1.18)

We first find

If we now add and subtract f(x)g(x + 1) on the right-hand side of the previous expression, then we obtain

Substituting g(x + 1) = g(x) + Δg(x) into the preceding expression yields Equation 1.18.

(1.19)

To see this, let

(1.20)

Here

(1.21)

We simply set

(1.22)

Set

(1.23)

We first find

Then from the binomial expansion formula (Eq. 1.14) applied to (x + 1)ⁿ, we have

or Equation 1.23.

Given the real-valued function f(x), we can, via the difference operator Δ, define a new function Δf(x). If we apply the operator Δ to this new function Δf(x), then we obtain the second difference of f(x) as the difference of the first difference or

(1.24)

Similarly, the third difference of f(x), which is the difference of the second difference, is

(1.25)

In general, the nth difference of f(x), which is the difference of the (n – 1)st difference of f(x), is

(1.26)

EXAMPLE 1.4 Given the real-valued function y = f(x) = x³ + 2x², find Δ³f(x). First,

Then, via the binomial expansion formula,

Next,

Finally,

The preceding example problem serves as a nice lead-in to the following result:

(1.27)

By property no. 8, Δ operating on xⁿ renders a finite number of terms with n – 1 as the highest power of x. Applying this observation to Equation 1.27 enables us to conclude that Δ operating on a polynomial of degree n results in a polynomial of degree n – 1. In a similar vein, Δ²f(x) will be a polynomial of degree n – 2, and Δⁿf(x) will thus be a polynomial of degree 0 (i.e., a constant). Moreover, for p > n, Δ^p applied to a constant must be zero.

1.5 THE NUMBER e

Let us consider the sequence (an ordered countable set of numbers not necessarily all different) x₁, x₂, …, x_n, …, where

(1.28)

If we expand the right-hand side of Equation 1.28 by the binomial formula (Eq. 1.14), then

(1.29)

Suppose we now replace n by n + 1 in Equation 1.28 so as to obtain

(1.30)

Again using the binomial expansion formula,

(1.31)

A term-by-term comparison of Equations 1.29 and 1.31 reveals that x_n+1 is always larger than x_n. In fact, Equation 1.31 has one more term than Equation 1.29. Hence, x_n+1 > x_n; that is, the sequence of values specified by Equation 1.28 is strictly monotonically increasing.

Next, looking to the expansion of x_n (Eq. 1.29), we see that

Since y_n is a geometric progression (the common ratio r = 1/2), we have

Hence, Equation 1.28 is bounded from above. And since any monotone bounded sequence has a limit, we can denote the limit of Equation 1.28 as

(1.28.1)

To five decimal places, e = 2.71828.

1.6 THE NATURAL LOGARITHM

We may define the natural logarithm of x, for positive x, as

(1.32)

(see Fig. 1.1). For x = 1, obviously ln 1 = 0; and for x < 1,

Given F(x) = ln x, it follows that F′(x) = dlnx/dx = 1/x.

Looking to the graph of the logarithmic function y = ln x (Fig. 1.2a), we see that ln x is continuous, single valued, and monotonically increasing with dy/dx = 1/x > 0, while d²y/dx² = –1/x²<0. Since ln 1 = 0, the curve passes through the point (1,0). Moreover, ln x → + ∞ as x→ + ∞; ln x = –ln 1/x→ –∞ as x→0+.

Figure 1.1 The natural logarithm of x.

Figure 1.2 (a) Logarithmic function and (b) Exponential function.

1.7 THE EXPONENTIAL FUNCTION

Given the logarithmic function y = ln x, if x=e. then ln e = 1 (Fig. 1.2a). Hence, the value of x for which ln x = 1 is e . Also, ln eⁿ=n ln e=n. Thus, the number whose natural logarithm is n is eⁿ so that the anti-natural logarithm of n is eⁿ or the antilogarithm is the inverse of the logarithm.

This said, given the logarithmic function y=ln x, its inverse function is

or

(1.33)

where x > 0 for y > 1, x = 0 for y = 1, and x < 0 for y < 1.

Thus, there exists a one-to-one correspondence between the sets Y = {y| y > 0} and X = {x| – ∞ < x < + ∞}. In this regard, we can define y (> 0) as a real-valued function of x for – ∞ < x + ∞. Hence,

(1.34)

is equivalent to Equation 1.33 and is called the exponential function. So with the logarithmic function continuous, single valued, and monotonically increasing, it follows that the exponential function, its inverse, exists and has the same exact properties. In sum,

Hence, e^x, defined for all real x, is that positive number y whose natural logarithm is x (Fig. 1.2b).

Some useful relationships between the exponential function and the (natural) logarithmic function are:

Moreover,

if ln y = ln a + w(ln r – ln s), then

if ln y = a + w(r – s), then

and if ln y = a + w(ln r – s), then

1.8 EXPONENTIAL AND LOGARITHMIC FUNCTIONS: ANOTHER LOOK

For the exponential function (Eq. 1.34), e served as the (fixed) base of this expression. Moreover, the unique inverse of Equation 1.34 is the logarithmic function y = ln x, where it is to be implicitly understood that “y is the natural logarithm of x to the base e.” However, other bases can be used.

Specially, let us alternatively specify an exponential function of x as

(1.35)

where b is the (fixed) base of the function. The base b will be taken to be a number greater than unity (since any positive number (y) can be expressed as a power (x) of a given number (b) greater than unity). Hence, Equation 1.35 is a continuous single-valued function, which is monotonically increasing for –∞ < x < + ∞ (Fig. 1.3a).

Since Equation 1.35 is continuous and single valued, it has a unique inverse called the logarithmic function

(1.36)

read “x is the logarithm of y to the base b“ (Fig. 1.3b). In this regard, a number x is said to be the logarithm of a positive real number y to a given base b if x is the power to which b must be raised in order to obtain y. Hence,

(Thus, log₅25 = 2 since 5² = 25; and log₂8 = 3 since 2³=8.)

Figure 1.3 (a) Exponential function (fixed base b) and (b) Logarithmic function (fixed base b).

Clearly a logarithm is simply a fancy way to write an exponent. Note that log_by is not defined for negative values of y or zero; that is, if 0 < y < 1, then log_by < 0; if y = 1, then log_b1 = 0; and if y > 1, then log_by > 0.

Some useful properties of logarithms are

Also, some useful differentiation formulas are

1.9 CHANGE OF BASE OF A LOGARITHM

Our goal is to develop a method for transforming the logarithm of x to the base b to the logarithm of x to the base a. To this end, we know from the preceding discussion of logarithms that y=log_ax is equivalent to x=a^y. Let us now take the logarithm of this latter expression with respect to the base b; that is,

or

(1.37)

Here Equation 1.37 will be termed our Change of Base Rule: the logarithm of x to the base a is the logarithm of x to the base b divided by logarithm of a to the base b.

It is well known that a common logarithm is taken to the base 10 (log₁₀100=2 since 10² = 100), while a natural logarithm, as defined earlier, is taken to the base e = 2.71828 (ln 20.08554 = 3 since e³ = 20.08554). We may employ Equation 1.37 to facilitate the conversion between them, that is,

1.10 THE ARITHMETIC (NATURAL) SCALE VERSUS THE LOGARITHMIC SCALE

Suppose the observations x_i, i, = 1,2, …, on a variable Xare measured relative to some specific scale and appear as points on the X-axis, with distances along this axis taken from some specific base or reference point. Now:

Looking to Figure 1.4, we see that on an arithmetic scale, either (i) the points appear at equal distances from each other (scale a) or (ii) the points appear at increasing distances from each other (scale b).

But as Figure 1.5 reveals, (i) taking the base 10 logarithms of the values on arithmetic scale a of Figure 1.4 produces a sequence of points exhibiting decreasing distances from each other (scale a′), or (ii) taking base 10 logarithms of the values on arithmetic scale b of Figure 1.4 renders a sequence of points that are located at equal distances from each other (scale b′).

Figure 1.4 Arithmetic scale.

Figure 1.5 Logarithmic scale.

If on an arithmetic scale a sequence of X values exhibits equal point-to-point decreases (e.g., consider 50, 40, 30, 20, 10), then the corresponding base 10 logarithmic scale displays values at increasing distances (to the left). And if the X values decrease by a fixed percentage on an arithmetic scale (e.g., for a 20% decrease we get 50, 40, 32, 25.6, 20.48), then the corresponding base 10 logarithmic values display equal point-to-point distances. (The reader is asked to verify these assertions for the given sets of arithmetic values.)

On the basis of the preceding discussion, it is evident that equal point-to-point distances on an arithmetic scale indicate equal absolute changes in a variable X; but equal point-to-point distances on a (base 10) logarithmic scale reflect equal proportional or percentage changes in X. For instance, if x₁, x₂, and x₃ are values of X plotted at equal distances on an arithmetic scale, then X increases by equal absolute amounts since x₃ – x₂ = x₂ – x₁. However, if the (base 10) logarithms of these X values are plotted at equal distances on a logarithmic scale, then X increases by equal proportional amounts since log₁₀x₃ – log₁₀x₂ = log₁₀x₂ – log₁₀x₁ or log₁₀(x₃/x₂) – log₁₀(x₂/x₁) or x₃/x₂ = x₂/x₁.

Suppose that instead of dealing with a single variable X, we introduce a second variable Y and posit a functional relationship between them of the form y = f(x), where f is a rule or law of correspondence (i.e., a mapping), which associates with each admissible value x of X a unique admissible value y of Y. Then in terms of our measurement scales, we note that:

1.11 COMPOUND INTEREST ARITHMETIC

Suppose a principal amount of $100.00 is invested and accumulates at a compound interest rate of 5% per year and interest is declared yearly. Then the following time profile of accumulation emerges:

In general, after t time periods or years, the accumulated amount at compound interest with annual compounding is

(1.38)

where P is the principal invested, 100r% is the yearly interest rate, and t indexes time in years. (In the preceding example, P = $100.00 and r=0.05.) So for, say, t = 10, A₁₀ = 100(1+0.05)¹⁰ = 100(1.62889) = $162.889. As this example problem reveals, the nature of compound interest is that, over the entire investment period, the interest itself earns interest.

What if interest is added twice a year rather than just once at the end of each year? Since the yearly interest rate is 5%, it follows that the half-yearly rate must be 2.5% so that 2.5% is added in each first half year and 2.5% is added in each second half year. So if a principal of $100.00 is invested and accumulates at a compound interest rate of 2.5% per half year and interest is declared at the end of each half year, then the revised time profile is:

To summarize, if interest is declared half-yearly, P is the principal, and the yearly interest rate is 100r%, then after t years the accumulated amount is

Again taking t = 10, A_2,10 = 100(1 + 0.025)²⁰ = $163.86144. A comparison of A₁₀ = $162.89 with A_2,10=$163.86 reveals that the more frequently interest is added, the larger is the accumulated amount at the end of a given period. In general, the accumulated amount at compound interest with interest declared j times a year is

(1.39)

A moment’s reflection concerning the structure of Equation 1.39 reveals that investment growth over time behaves as a geometric progression; that is, each amount is a fixed multiple (1 + (r/j))^j of the previous period’s amount. That is, the sequence of terms of this geometric progression is:

Hence, the growth process represented by Equation 1.39 can be expressed as the exponential function

(1.40)

and is referred to as a compound interest growth curve (alternatively called a geometric or exponential growth curve). Transforming to logarithms gives

(1.41)

Clearly this semilogarithmic expression plots as a straight line with vertical intercept log₁₀P and (constant) slope log₁₀(1 + (r/j)). Obviously the magnitude of the slope depends upon r and j. In this regard, r/j is the proportionate rate of change in A_j,t per unit period of time (i.e., per year if j = 1, per half year if j = 2, per quarter if j = 4). Note that the independent variable on the right-hand side of Equation 1.41 is jt; it represents the total number of subperiods j within a year times the number of years t.

For instance, if j = 4, then jt = 4t represents the total number of quarters spanned by the entire accumulation period; that is, if t = 1, 4t = 4 spans one year; if t = 2, 4t = 8 spans a two-year time interval.

A special case of Equation 1.41 is, from Equation 1.38,

(1.41.1)

where r is the proportionate rate of growth in A_t per unit of time.

Given Equation 1.39, let us assume that j increases without limit or, equivalently, that the compounding or conversion periods become shorter and shorter. In this instance the term (1 + (r/j)) in Equation 1.39 is replaced by e, and we consequently have what is termed the case of continuous compounding or continuous conversion.

To see this, let us rewrite Equation 1.39 as

where n = j/r. Now, as the number of compounding or conversion periods j → + ∞, it follows that n → + ∞. Hence, via Equation 1.28.1,

(1.42)

Here Equation 1.42 depicts a natural exponential growth curve and represents the accumulated amount at the end of t years if the principal (P) grows at an exponential rate of 100r% per year or is compounded continuously at 100r% per year. For instance, if we again take P = $100, r = 0.05, and t = 10, then A = 100e^0.05(10)=$164.872.

An assortment of points concerning Equation 1.42 merits our attention. First, the variable t in Equation 1.39 is discontinuous since interest is declared at specific (discrete) intervals over the investment period. However, as j → +∞ and interest is declared with increasing frequency, t tends to become a continuous variable in Equation 1.42. Second, it is instructive to view the term in Equation 1.42 as the yearly accumulated amount of $1.00 invested when interest is compounded at 100% per annum and declared n times over the year. Then as n increases without bound,

So as compound interest is declared more and more frequently, the $1.00 invested at 100% interest approaches $e at the end of a year. Third, transforming Equation 1.42 to logarithms yields

(1.43)

which plots as a straight line with vertical intercept ln P and (constant) slope r. Finally, if an amount is compounded yearly at 100r% (e.g., see Eq. 1.38) and that same amount is compounded continuously at 100g% and 100r% and 100g% are equivalent interest rates, then obviously g = ln(1 + r).

¹ Here r! is called r factorial and is calculated as r! = r(r – 1)(r – 2)···3·2·1 with 0! ≡ 1.

² It is instructive to view the derivation of this expression in an alternative light. To this end, let us write Equation 1.39 as

where x=r/j. Then applying the binomial formula 1.12 to the term in square brackets yields

As j → +∞, it follows that x → 0 so that

via Equation 1.28.1.

2

FUNDAMENTALS OF GROWTH

2.1 TIME SERIES DATA

We may view a time series as a set of observations Y_t, t = 0, 1, 2, …, n, on a variable Y that are indexed in order of time; that is, the Y_t’s are measured at different time points or time intervals. Since t is the time index, if our data series starts in, for instance, 1980 and we have observations going to 2005 in one-year intervals, then we can assign a sequence of numbers to order the data points. In this regard, let us designate 1980 as the origin and assign it the value “0.” Then 1981 is assigned the value 1, 1982 is given the value 2, and so on. Hence, we end up with the sequence of numbers 0, 1, 2, …, 25 as the set of observations on the time variable t. The convenience of this numerical assignment scheme for representing a sequence of years (or weeks, months, etc.) will become evident later on.

2.2 RELATIVE AND AVERAGE RATES OF CHANGE

Given the time series Y_t, t = 0, 1, n, let us define the relative rate of change in Y between periods t – 1 and t as

(2.1)

TABLE 2.1 Relative Growth Rates in Y and Growth Relatives

If Y₀ denotes the value of Y at the beginning of period 1 and Y_t represents the value of Y at the end of period t, t = 1, …, n, then the sequence of relative rates of growth over the entire time span appears in Table 2.1.

Moreover, the ratios Y_t/Y_t–1 (=R_t + 1), t = 1, 2, …, n, are termed growth relatives and, from Table 2.1, appear as

(2.2)

EXAMPLE 2.1 Given the Y values appearing in Table 2.2 , determine the sequence of growth relatives along with the period-to-period relative rates of growth in Y. Let Y₀ = 200. For instance, Y experiences almost a 17% rate of growth between periods 3 and 4 since R₄ = 0.1666 or 100(0.1666)% = 16.66%.

What is average rate of growth in Y over the four periods given in the preceding example problem? At first blush one might be tempted to take the simple arithmetic average of the R_t’s so as to obtain

or 17.13%. Interestingly enough, this calculation would be misleading. The arithmetic average is the incorrect average to be used when it comes to finding the average rate of growth; the appropriate average rate of growth over the four periods is given by the geometric mean

(2.3)

TABLE 2.2 Growth Relatives and Relative Rates of Growth

EXAMPLE 2.2 Given the Y values presented in Table 2.2, find the appropriate average rate of growth or average percentage change in Y over the indicated four periods. We now determine

or

Hence, the four-period average rate of growth in the Y series is about 15.02%; that is, between periods one and four, Y increased in value from Y₀ = 200 to Y₄ = 350, an average increase of 15.02%.

Let us rewrite Equation 1.38 or A_t = P(1 + r)^t as

Then

so that

(2.5)

Hence, the average rate of growth found by taking the geometric mean (Eq. 2.3) is simply the (annual) rate of interest in the compound interest formula (Eq. 1.38); that is, average growth over the four-period span is equivalent to compound interest growth. (So if the principal is 200 and r = 0.150153, then Y₄ = 200 (1.150153)⁴ = 200 (1.749998) = 350 as expected.)

EXAMPLE 2.3 Suppose Y assumes the values 5, 7, 10, 11, and 20%. To average this set of percents, let us find

Note that GM < = (5 + 7 + 10 + 11 + 20)/5 = 53/5 = 10.6 since the geometric mean is not so severely affected by extremes as the arithmetic mean.

EXAMPLE 2.4 Suppose the value of Y in 1990 was Y₁₉₉₀= 1000 and its value in 2000 was Y₂₀₀₀ = 2500. What is the average rate of growth or average percentage increase in Y between 1990 and 2000 (here n = 11)? We need to find

So between 1990 and 2000, Y increased by approximately 9.6% on average.

In view of Example 2.4, let us modify Equation 2.3 slightly to consider the instance when we are faced with finding the average rate of growth in Y over the time span t = 1, …, n. Then

(2.3.1)

In a similar vein, the compound interest rate of growth is determined by compounding over t – 1 periods; that is, from Y_t = Y₁(1 + r)^t–1, Equation 2.5 becomes

(2.5.1)

2.3 ANNUAL RATES OF CHANGE

2.3.1 Simple Rates of Change

We noted in Equation 2.1 earlier that the relative rate of change in Y between periods t – 1 and t is

or, in percent terms,

In this regard, the percent change from a year ago in Y is the percent change from the same period in the previous year. Hence, the percent change from a year ago in Y between quarter t – 4 and the current quarter t is

(2.6)

while the percent change from a year ago in Y between month t – 12 and the current month t is

(2.7)

Next, for consecutive quarters, the percent change at an annual rate in Y between the previous quarter t – 1 and the current quarter t is the quarterly percentage change multiplied by 4 or

(2.8)

and, for consecutive months, the percent change at an annual rate in Y between the previous month t – 1 and the current month t is the monthly percentage change multiplied by 12 or

(2.9)

2.3.2 Compounded Rates of Change

Let us now compound the simple quarterly or monthly percent changes so as to express them as annual growth rates. In this regard, by the compounded annual rate of change in a series, we mean its growth rate over an entire year if the same simple percent change continued for four quarters or 12 months. More specifically, let us define the compound annual rate of change in Y between the previous quarter t – 1 and the current quarter t as

(2.10)

while the compound annual rate of change in Y between the previous month t – 1 and the current month t is

(2.11)

For instance, we can easily rationalize Equation 2.10 as follows. Let us rewrite Equation 1.38 as

(2.12)

where Y_t is the value of Y in the current period (t), Y_t–1 is the value of Y in the previous period (t – 1), and 100r % is the (compound) annual rate of change in Y. Given that r represents annual compounding, Equation 2.12 becomes, for quarterly data,

Then solving this expression for r enables us to write the compound annual rate of change in Y between the current quarter t and the previous quarter t – 1 as

In a similar vein, for monthly data, Equation 2.12 becomes

where again 100r % is the (compound) annual rate of change. Then the compound annual rate of change in Y between the current month t and the previous month t – 1 is

EXAMPLE 2.5 For quarterly observations on Y, suppose Y_t–1 = 100 (the first quarter, say, begins with Y = 100) and Y_t, = 105 (the second quarter begins with Y = 105). Then, from Equation 2.10,

Hence, the compound annual rate of change in Y between the current quarter and the previous quarter is 21.5515%; that is, 21.551% is the growth rate over the entire year if the same simple percent (21.551%) continued over all four quarters. Hence, annual growth (from the beginning of the first quarter through the end of the fourth quarter) at 21.551% yields

But this is the accumulated amount we would expect at the end of a year if the Y series started at Y₀ = 100 and grew at the compound annual rate of 21.551% from quarter to quarter:

For year-to-date calculations on quarterly data, the annualized year-to-q percent change is obtained from the formula

(2.10.1)

where Y_t–1,4 is the value of Y in the fourth quarter of year t – 1, q is the number of quarters under consideration for year t, and Y_t,q is the value of Y in the qth quarter of year t. And for year-to-date calculations for monthly data, the annualized year-to-m percent change is obtained via the formula

(2.11.1)

where Y_t–1,12 is the value of Y in December of year t – 1, m is the number of the month under consideration for year t, and Y_t,m is the Y value in the mth month of year t.

EXAMPLE 2.6 Table 2.3 presents the monthly values of a variable Y from the last month (December) of 2008 up through the first eight months of 2009. As column three of this table reveals, Y grew by 14.93% during the first six months of 2009. Then in July and August of 2009, Y grew by 3.12% and 2.78% respectively. Is Y growth in July and August above or below the rate exhibited during the first six months of 2009?

To answer this question, we need to annualize these growth rates; that is, the growth rates need to be adjusted to reflect the amount Y would have changed over the course of a year if it had continued to grow at the given rate. We note first that the July growth rate is calculated, via Equation 2.1, as

where 774.7 is the value of Y in July and 751.3 is the Y value for June. Similarly, the August growth rate is calculated as

TABLE 2.3 Monthly Percent Changes in Y_t

Also, the simple percent change from December 2008 to June 2009 is

(Here we are applying a slight modification of Equation 2.1 or

(2.1.1)

where Y_t–m is the level of Y recorded m periods prior to period t. For this calculation, m = 6.)

In order to compare these three growth rates, we need to annualize them. An application of Equation 2.11 for July gives

(see column four of Table 2.3). Thus, 44.49% is the amount Y would have increased for the entire year if it had grown at the monthly rate of 3.12% for all 12 months. A second application of this formula for August yields

Here 38.88% is the amount Y would have increased for the entire year if it had expanded at the monthly rate of 2.78% over the whole 12-month period. To obtain the annualized growth rate from December 2008 to June 2009, let us use Equation 2.11.1 to find

This year-to-m(=6) rate depicts the amount Y would have increased for the whole year if it had continued to grow at the pace experienced between January and June. On the basis of these annualized growth rates, we see that growth in Y for both the months of July and August is above the rate experienced in the first six months of 2009 (even though the column three entries in Table 2.3 could possibly lead one to conclude otherwise).

2.3.3 Comparing Two Time Series: Indexing Data to a Common Starting Point

In order to compare the performance of two time series data sets, say X_t and Y_t, we need to index the data to a common starting point. Hence, the initial values of X and Y must be set equal to each other so that differences over time between the X and Y variables can be highlighted. Once indexing to a common starting point is accomplished, we can readily determine their rates of growth, especially when the X_t and Y_t series are stated in different units. For instance, suppose Table 2.4 houses the values of the two time series data sets X_t and Y_t from 1998 to 2008. The time paths of these variables are depicted in Figure 2.1.

To index a set of time series data values, the observations must be made equal to each other at some given starting date. Typically, this value is 100. From this initial point onwards, every value is normalized to the starting value so as to preserve the same percentage value as in the original or indexed series. Subsequent values are then determined so that percentage changes in the indexed time series are coincident with those for the unindexed series. In sum, indexing involves modifying two (or more) time series data sets so that the resulting or indexed series start at the same value and change at the same rate as the unmodified or unindexed series.

So for data series X_t in Table 2.4, the indexed X_t’s are calculated as

(2.2.1)

where denotes the indexed value of X_t and X₀ is the initial value of X. For the Y data set, the indexed Y_t’s are determined as

(2.2.2)

with denoting the indexed value of Y and Y₀ representing the initial value of Y. The indexed values for both variables X and Y appear in Table 2.4 (here X₀ = 57 and Y₀ = 420).

TABLE 2.4 Original and Indexed Time Series Data Sets X_t, Y_t

Figure 2.1 Time series observations on X, Y.

What sort of information can be garnered from the indexed X and Y time series data? Between 1998 and 1999, the unindexed variable X_t increased from 57 to 60 or 5.26%, while the indexed variable also increased by 5.26% (i.e., by 105.26 – 100 = 5.26). For this same time period, the unindexed variable Y_t increased from 420 to 424 or by 0.95%, while its indexed counterpart also increased by 0.95% (i.e., by 100.95 – 100 = 0.95).

What about the change in the X series from 1998 to, say, 2002? Here X_t increases by 28.07% (128.07 – 100 = 28.07); and between 1998 and 2002, the Y series increased by 2.86% (102.86 – 100 = 2.86). Additionally, for the entire 1998–2008 time span, the X data series grew by 64.91% (164.91 – 100 = 64.91), while the Y data series grew by only 7.62% (or 107.62 – 100 = 7.62) (Fig. 2.2).

Figure 2.2 Time series observation on , .

2.4 DISCRETE VERSUS CONTINUOUS GROWTH

Suppose time t is measured in discrete units or intervals (e.g., the observations on some variable are made yearly, monthly, etc.). Then a constant growth series for a variable Y can be modeled as

(2.13)

where r is the constant proportionate rate of growth in Y per unit of time; that is, from Equation 2.13, the period-to-period or relative growth rate (RGR) is

the constant proportionate (or compound interest) rate of growth.

Transforming Equation 2.13 to logarithms gives

(2.13.1)

where the slope of this linear equation is log ₁₀(1 + r). Clearly this slope coefficient can be estimated (via ordinary least squares (OLS)) and denoted as

Then to find , set 1 + = 10^b or

(2.14)

If t is measured in one-year increments, then the estimated rate of growth in Y is 100% per annum. Hence, the quarterly rate of growth must be

or

(2.15)