economics paper 2: quantitative methods- ii (statistical

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ECONOMICS

Paper 2: Quantitative Methods- II (Statistical Methods)

Module 34: Exponential non-linear equation

Subject ECONOMICS

Paper No and Title 2: Quantitative Methods- II (Statistical Methods)

Module No and Title 34: Exponential non-linear equation

Module Tag ECO_P2_M34

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ECONOMICS



TABLE OF CONTENTS

1. Learning Outcomes

2. Introduction

3. Growth Functions

4. Growth rate

5. Forms of exponential function

6. Properties and restrictions

7. Comparative functions

8. Estimation of an exponential function

9. Summary

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1. Learning Outcomes

After studying this module, you shall be able to

1. Know about the basic structure of an Exponential Function;

2. Understand how growth rates are calculated;

3. Understand the properties of Exponential Functions.

4. Estimate an Exponential Function

2. Introduction

Amongst the class of non-linear functions we have:

i) Inverse function;

ii) Quadratic function;

iii) Cubic function;

iv) Gompertz function;

v) Logistic function; and

vi) Exponential function.

In this module we are considering the properties, path and estimation of the Exponential

function. We would also learn how to estimate the compound growth rate with the help of

the Exponential function.

This the most useful function for doing growth analysis. Various economic variable like

GDP, Money Supply, consumption, etc, need to be measured in terms of their growth rates.

Here the exponential function comes in handy.

3. Growth Function

The general form of the exponential function is:

The variable ‘x’ has appears as a power term.

Similarly,

f(x) = bx ± c

could also be an exponential function. This is because it is derived from

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f(x) = ebx ± c

could also be re-written by a logarithmic transformation1:

f(Lx) = bx ± c

This is known as log-linearization. Here L stands for Log to the base ‘e’.

Exponential functions are growth functions. They are characterized by the fact that they

can capture the growth rate directly.

where constant stands for the intercept term.

When the base is the number e ≈ 2.71828...:

This is known as a "natural exponential function" and can be expressed as:

The exponential function model has a definite structure and purpose. Generally it is defined

over time:

Yt = ea+bt

The structure is that any variable is regressed on time. Usually it has n intercept that tells

us the minimum level at which the phenomenon being measured starts. The variable Yt

could be income, prices, consumption, etc.

The purpose is to enable the determination of the growth rate, usually treated as the

exponential growth rate. The relationship between Yt and t is that Yt grows in proportion

to ‘t’ that is time. The change over time is proportional.

4. Growth Rate

How does the exponential function help in knowing the growth rate2?

We consider

Yt = ea+bT

And take the natural log of both sides. This is known as log-linearizing.

LYt = a + b*T …..(1)

at point ‘T’3.

LY (t-1) = a + b*(T-1) …. (2)

at point‘t-1’.

1 Note that here log refers to natural log and log to the base 10. 2 Animation of exponential function

https://en.wikipedia.org/wiki/Exponential_function#/media/File:Animation_of_exponential_function.gif 3 Note that here t is not a subscript. It is the time variable ‘T’ that is multiplied by ‘b’.

https://en.wikipedia.org/wiki/Exponential_function#/media/File:Animation_of_exponential_function.gif

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If we subtract Eq. (2) from Eq. 1:

LYt - LY (t-1) = b*t – b*(T-1) …. (3)

LYt - LY (t-1) = b*1 or b …. (4)

LYt / LY (t-1) = b …. (5)

This means that the variable Yt is growing over time in such a manner that the relative

change in Yt over time is equal to a constant ‘b’. It is relative because it is Log Yt that is

changing over time. It is not absolute change in Yt. This ‘b’ is the constant growth rate.

b = ∆Yt/∆t …. (6)

∆t= 1and ∆Yt = L(Yt/Yt-1) … (7)

That is, a unit change in time (T), say, one year. Thus, growth in Yt in one year is nothing

but the growth rate of Yt per annum.

This can also be derived as:

Y t=Y0 (1+r)T …. (8)

Taking log of both sides:

LYt=LY0+ T[L(1+r)] …. (9)

LYt=LY0+ T[L(1+r)] …. (10)

We can re-write (10) as:

Yt= β0 + β1T …… (11)

Where

β0= LY0 and β1= [L(1+r)] …… (12)

Now;

Anti-log β1= Antilog [L(1+r)] …… (13)

Anti-log β1= 1+r ……. (14)

Anti-log β1-1= r ……. (15)

Thus, the growth rate of Yt is ‘r’. Here, there are two growth rates. The first is β1 which is

the instantaneous growth rate. On the other hand, Anti-log β1-1is the annual compound

growth rate or ACGR.

5. Forms of Exponential Functions

The exponential function ex can be characterized in a variety of equivalent ways. In

particular it may be defined by the following

Using an alternate definition for the exponential function leads to the same result when

expanded as a Taylor series. Less commonly, ex is defined as the solution y to the equation

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It is also the following limit:

An exponential function will look like this:

Figure 1: Shape of an Exponential Function: f(x)

We plot a growth function f(x):

Y = f(x) = 2x

As we go towards the negative x-axis away from the origin y=f(x) keeps on halving.

Therefore, the exponential function starts as a very small fractional number. It almost

touches the x-axis. In the beginning ‘y’ is extremely small, such that ‘y’ is indistinguishable

from "y = 0", which is the x-axis. Once ‘y’ starts growing faster it accelerates very fast.

The term "exponential growth" f(x) implies "starting slow, but then growing very fast”.

The function f(x) doubles when ‘x’ increases at a constant rate. That is, when x was

increases by 1 unit of time, ‘y’ increases to twice what it had been. The increments to ‘x’

must be fixed – one day, one week, one month and one year. For each increment‘t’, for

instance, the urban population ‘y’ during each decade will double. In the beginning of last

century population hardly ever grew in India. Now, it is growing at an exponential rate.

Between 1 and 2 the growth is 2; between 2 and 3 it is 4; between 3 and 4 it is 8, and so on

(Figure 1).

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6. Properties and Restrictions

Figure 2: Nature of Exponential Function

a. Asymptotic Properties:

Path of the exponential function from - ∞ to +∞

If x= 0

ʄ (0) = (b)0= 1

If x=1

ʄ (1) = (b)1= b

When x approaches the origin from - ∞

x →0, ʄ (0) →1

The coordinates of the function (x, y) are (0, 1).

When x approaches the origin from + ∞

x →1, ʄ (1) →b or e

The coordinates of the function (x, y) are (1, e). These coordinates are shown in Figure 2

above.

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If 0<b<1 the exponential function will be downward sloping. If b>1 then it would be

upward sloping.

Figure 3: Decaying function

The above function is based on

g (x) = (0.5)x

Since b< 1 the function is decaying. It moves downwards from left to right (Figure 3).

b. Restrictions:

There are some restrictions on the parameters that influence the exponential function. First,

there is a restriction on extreme values of ‘b’. We must avoid 1 and 0 since the log of 0

and 1 are constant functions and won’t have many of the same properties that general

exponential functions have.

Thus,

ʄ (x) = (0)x= 0

and

ʄ (x) = (1)x= 1

It is necessary to avoid negative numbers for ‘b’. This may lead to an imaginary numbers.

For instance if we consider and x= ½

ʄ (x) = (-4)1/2 = √ (-4)

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7. Comparative Functions

Before estimating first of all we must identify which kind of function is appropriate. There

could be at least three possibilities.

i) Linear;

ii) Polynomial; and

iii) Exponential.

The linear function is:

y=5x

The polynomial is

y = x2+ 3x

And

y = 1.5x

The slope of a linear function is constant. The slope of a polynomial function is gradual.

The slope of an exponential function is sudden in both directions: when x increases and

when x decreases. At x=12 the height of the quadratic is more than the exponential

function. At x= 13 the height of the exponential function overtakes the quadratic. When

x=9 even the liner function’s height is greater than the exponential function. But when x=

10 the exponential curve suddenly increases in height. These relationship are given in

Figure 4.

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Figure 4: Comparative functions

8. Estimation of an Exponential Function

Now we shall estimate an empirical exponential function and estimate the growth rate. As

we have seen there are two growth rates:

i) Instantaneous Growth Rate

ii) Annual Compound Growth Rate

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Figure 5: India’s GDP Growth

Using a semi-log equation we have estimated a growth function.

GDP = e a+bT

Log-linearizing and adding an error term

Log GDPt = a + bT + Ut

Figure 5 shows how a good estimate of the growth rate (path) will depict a close

relationship between the actual and predicted GDP.

The instantaneous growth rate is at a point of time, whereas the annual compound growth

rate is for the entire length of time. The Instantaneous Growth Rate would be the slope at

each point on the curve. The Annual Compound Growth Rate tells us how a variable that

started so low swells into such a large amount.

i) Instantaneous Growth Rate: b= 0.1325 x 100 = 13.25 growth rate.

ii) Annual Compound Growth Rate: Antilog (b) -1 = 14.16

The growth rate obtains from the estimate from the regression equation given below:

Log GDPt = -234.37 + (0.13249)*T + Ut

t-stat (-49.48) (56.14)

Table 1 contains the regression output for the above equation. The t-statistics show that

both the intercept and slope (b= growth rate) are highly significant. The R2shows that the

equation is a good fit.

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SUMMARY OUTPUT Dependent Variable: LGDP

Regression Statistics

Multiple R 0.9979446

R Square

0.9958934

3

Adjusted R

Square

0.9955775

4

Standard Error

0.0394863

3

Observations 15

ANOVA

df SS MS F

Significanc

e F

Regression 1

4.9155376

9

4.91553769

5

3152.6618

1 6.6836E-17

Residual 13

0.0202692

2

0.00155917

1

Total 14

4.9358069

1

Coefficient

s

Standard

Error t Stat P-value

Intercept -234.37193

4.7360489

8 -49.4868043 3.4272E-16

Year

0.1324971

4

0.0023597

6

56.1485691

2 6.6836E-17

Table 1: India’s GDP growth equation

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9. Summary

We have discussed at length the nature, shape and properties of an Exponential Function.

1. Now we know about the basic structure of an Exponential Function which is

2. There are two types of growth rates: Instantaneous and Annual Compound Growth

rates. We have understood how growth rates are calculated.

3. The instantaneous growth rate is at a point of time, whereas the annual compound

growth rate is for the entire length of time. The Instantaneous Growth Rate would

be the slope at each point on the curve. The Annual Compound Growth Rate tells

us how a variable that started so low swells into such a large amount.

4. We have understood the properties of Exponential Functions. The path it follows

is:

When x approaches the origin from - ∞

x →0, ʄ (0) →1

The coordinates of the function (x, y) are (0, 1).

When x approaches the origin from + ∞

x →1, ʄ (1) →b or e

The coordinates of the function (x, y) are (1, e).

5. With the help of a semi-log equation we have estimated the growth rates of India

GDP growth path that is based on an Exponential Function.

economics paper 2: quantitative methods- ii (statistical

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