Basic Mathematics for Economists - Rosser - Chapter 14 pps

14 Exponential functions, continuousgrowth and differential equations Learning objectives After completing this chapter students should be able to: • Use the exponential function and nat

14 Exponential functions, continuous

growth and differential equations

Learning objectives

After completing this chapter students should be able to:

• Use the exponential function and natural logarithms to derive the final sum, initialsum and growth rate when continuous growth takes place

• Compare and contrast continuous and discrete growth rates

• Set up and solve linear first-order differential equations

• Use differential equation solutions to predict values in basic market and economic models

macro-• Comment on the stability of economic models where growth is continuous

14.1 Continuous growth and the exponential function

InChapter 7, growth was treated as a process taking place at discrete time intervals In thischapter we shall analyse growth as a continuous process, but it is first necessary to understandthe concepts of exponential functions and natural logarithms The term ‘exponential function’

is usually used to describe the specific natural exponential function explained below However,

it can also be used to describe any function in the format

y = Ax where A is a constant and A > 1

This is known as an exponential function to base A When x increases in value this function obviously increases in value very rapidly if A is a number substantially greater than 1 On the other hand, the value of A x approaches zero if x takes on larger and larger negative values For all values of A it can be deduced from the general rules for exponents (explained in

Chapter 2) that A0= 1 and A1= A.

Example 14.1

Find the values of y = A x when A is 2 and x takes the following values:

(a) 0.5, (b) 1, (c) 3, (d) 10, (e) 0 , (f)−0.5, (g) −1, and (h) −3

The natural exponential function

In mathematics there is a special number which when used as a base for an exponentialfunction yields several useful results This number is

2.7182818 (to 7 dp)

and is usually represented by the letter ‘e’ You should be able to get this number on yourcalculator by entering 1 and then using the [ex] function key

To find ex for any value of x on a calculator the usual procedure is to enter the number (x)

and then press the [ex] function key To check that you can do this, try using your calculator

to obtain the following exponential values:

In economics, exponential functions to the base e are particularly useful for analysinggrowth rates This number, e, is also used as a base for natural logarithms, explained later

in Section 14.4 Although it has already been pointed out that, strictly speaking, the specific

function y = ex should be known as the ‘natural exponential function’, from now on weshall adopt the usual convention and refer to it simply as the ‘exponential function’

To understand how this rather awkward value for e is derived, we return to the methodused for calculating the value of an investment developed inChapter 7 You will recall that

the final value (F ) of an initial investment (A) deposited for t discrete time periods at an interest rate of i can be calculated from the formula

F = A(1 + i) t

If the interest rate is 100% then i= 1 and the formula becomes

F = A(1 + 1) t = A(2) t

Assume the initial sum invested A= 1 If interest is paid at the end of each year, then after

1 year the final sum will be

F1= (1 + 1)1= 2

InChapter 7it was also explained how interest paid monthly at the annual rate divided by 12will give a larger final return than this nominal annual rate because the interest credited each

month will be reinvested When the nominal annual rate of interest is 100% (i= 1) and theinitial sum invested is assumed to be 1, the final sum after 12 months invested at a monthlyinterest rate of 121(100%) will be

This result means that a sum A invested for one year at a nominal annual interest rate of

100% credited continuously will accumulate to the final sum of

to go to the 4th decimal place to find a difference between the two.) Continuous growthalso occurs in other variables relevant to economics, e.g population, the amount of naturalmaterials mined Other variables may continuously decline in value over time, e.g the stock

of a non-renewable natural resource

14.2 Accumulated final values after continuous growth

To derive a formula that will give the final sum accumulated after a period of continuousgrowth, we first assume that growth occurs at several discrete time intervals throughout

a year We also assume that A is the initial sum, r is the nominal annual rate of growth, n is

the number of times per year that increments are accumulated and y is the final value Using

the final sum formula developed inChapter 7, this means that after t years of growth the final

Growth becomes continuous as the number of times per year that increments in growth are

accumulated increases towards infinity When n→ ∞ then n

The final value of the population (in millions) is found by using the formula y = Ae rt and

substituting the given numbers: initial value A = 4.5; rate of growth r = 3% = 0.03; number

of time periods t = 15, giving

y = 4.5e 0.03(15) = 4.5e 0.45 = 4.5 × 1.5683122 = 7.0574048 million

Thus the predicted final population is 7,057,405

Example 14.3

An economy is forecast to grow continuously at an annual rate of 2.5% If its GNP is currentlye56 billion, what will the forecast for GNP be at the end of the third quarter the year afternext?

Solution

In this example: t = 1.75 years, r = 2.5 % = 0.025, A = 56 (e billion) Therefore, the

final value of GNP will be

y = Ae rt = 56e0.025(1.75)= 56e0.04375 = 58.504384

Thus the forecast for GNP ise58,504,384,000.

So far we have only considered positive growth, but the exponential function can also

be used to analyse continuous decay if the rate of decline is treated as a negative rate ofgrowth

Therefore, the river flow will shrink to 14.16 million gallons per day

Continuous and discrete growth rates compared

In Section 14.1 it was explained how interest at a rate of 100% credited continuously

through-out a year gives an annual equivalent rate of r = e−1 = 1.7182818 = 171.83%, a difference

of 71.83% However, in practice interest is usually credited at much lower annual rates Thismeans that the difference between the nominal and annual equivalent rates when interest iscredited continuously will be much smaller This is illustrated inTable 14.1for the case whenthe nominal annual rate of interest is 6%

These figures show that the annual equivalent rate when interest is credited continuously isthe same as that when interest is credited on a daily basis, if rounded to two decimal places,although there will be a slight difference if this rounding does not take place

Test Yourself, Exercise 14.1

1 A country’s population is currently 32 million and is growing continuously at anannual rate of 3.5% What will the population be in 20 years’ time if this rate ofgrowth persists?

2 A company launched a successful new product last year The current weekly saleslevel is 56,000 units If sales are expected to grow continually at an annual rate

of 12.5%, what will be the expected level of sales 36 weeks from now? (Assumethat 1 year is exactly 52 weeks.)

3 Current stocks of mineral M are 250 million tonnes If these stocks are continuallybeing used up at an annual rate of 9%, what amount of M will remain after 30years?

4 A renewable natural resource R will allow an estimated maximum consumptionrate of 200 million units per annum Current annual usage is 65 million units

If the annual level of usage grows continually at an annual rate of 7.5% will there

be sufficient R to satisfy annual demand after (a) 5 years, (b) 10 years, (c) 15 years,(d) 20 years?

5 Stocks of resource R are shrinking continually at an annual rate of 8.5% Howmuch will remain in 30 years’ time if current stocks are 725,000 units?

6 Ife25,000 is deposited in an account where interest is credited on a daily basis

that can be approximated to the continuous accumulation of interest at a nominalannual rate of 4.5%, what will the final sum be after five years?

14.3 Continuous growth rates and initial amounts

Derivation of continuous rates of growth

The growth rate r can simply be read off from the exponent of a continuous growth function

in the format y = Ae rt

To prove that this is the growth rate we can use calculus to derive the rate of change of thisexponential growth function

If variable y changes over time according to the function y = Ae rt then rate of change of

y with respect to t will be the derivative dy/dt However, it is not a straightforward exercise

to differentiate this function For the time being let us accept the result (explained below in

Section 14.4) that

if y= et thendy

dt = et

i.e the derivative of an exponential function is the function itself

Thus, using the chain rule,

when y = Ae rt

then dy

dt = rAe rt This derivative approximates to the absolute amount by which y increases when there is a one unit increment in time t, but when analysing growth rates we are usually interested in the proportional increase in y with respect to its original value The rate of growth is therefore dy

dt

y =rAert

Aert = r

Even though r is the instantaneous rate of growth at any given moment in time, it must be

expressed with reference to a time interval, usually a year in economic applications, e.g 4.5%per annum It is rather like saying that the slope of a curve is, say, 1.78 at point X A slope of1.78 means that height increases by 1.78 units for every 1 unit increase along the horizontalaxis, but at a single point on a curve there is no actual movement along either axis

Example 14.5

Owing to continuous improvements in technology and efficiency in production, an empirical

study found a factory’s output of product Q at any moment in time to be determined by the

function

Q= 40e0.03t

where t is the number of years from the base year in the empirical study and Q is the output

per year in tonnes What is the annual growth rate of production?

Solution

When the accumulated amount from continuous growth is expressed by a function in the

format y = Ae rt then the growth rate r can simply be read off from the function Thus when

Q= 40e0.03t

the rate of growth is

r = 0.03 = 3%

Initial amounts

What if you wished to find the initial amount A that would grow to a given final sum y after

t time periods at continuous growth rate r? Given the continuous growth final sum formula

y = Ae rt

then, by dividing both sides by ert, we can derive the initial sum formula

A = ye −rt

Example 14.6

A parent wants to ensure that their young child will have a fund of £35,000 to finance his/her

study at university, which is expected to commence in 12 years’ time They wish to do this

by investing a lump sum now How much will they need to invest if this investment can beexpected to grow continuously at an annual rate of 5%?

Test Yourself, Exercise 14.2

1 A statistician estimates that a country’s population N is growing continuously and

can be determined by the function

N = 3,620,000e 0.02t

where t is the number of years after 2000 What is the population growth rate?

Will population reach 10 million by the year 2050?

2 Assuming that oil stocks will continue to be depleted at the same continuous rate(in proportion to the amount remaining), the amount of oil remaining in an oilfield

(B), measured in barrels of oil, has been estimated as

B = 2,430,000,000e −0.09t

where t is the number of years after 2000 What proportion of the oil stock is

extracted each year? How much oil will remain by 2020?

3 An individual wants to ensure that in 15 years’ time, when they plan to retire, they

will have a pension fund of £240,000 They wish to achieve this by investing a

lump sum now, rather than making regular annual contributions If their investment

is expected to grow continuously at an annual rate of 4.5%, how much will theyneed to invest now?

4 The owner of an artificial lake, which has been created with the main aim ofmaking a commercial return from recreational fishing, has to decide how manyfish to stock the lake with Allowing for the natural rate of growth of the fishpopulation and the depletion caused by fishing, the number of fish in the lake isexpected to shrink continuously by 3.2% a year How many fish should the ownerstock the lake with if they wish to ensure that the fish population will still be 500

in 5 years’ time, given that it will not be viable to add more fish after the initialstock is introduced?

14.4 Natural logarithms

InChapter 2, we saw how logarithms to base 10 were defined and utilized in mathematical

problems You will recall that the logarithm of a number to base X is the power to which X

must be raised in order to equal that number Logarithms to the base ‘e’ have several usefulproperties and applications in mathematics These are known as ‘natural logarithms’, and theusual notation is ‘ln’ (as opposed to ‘log’ for logarithms to base 10)

As with values of the exponential function, natural logarithms can be found on a matical calculator Using the [LN] function key on your calculator, check that you can derivethe following values:

mathe-ln 1= 0

ln 2.6 = 0.9555114

ln 0.45 = −0.7985

The rules for using natural logarithms are the same as for logarithms to any other base Forexample, to multiply two numbers, their logarithms are added But how do you then transformthe sum of the logarithms back to a number, i.e what is the ‘antilog’ of a natural logarithm?

To answer this question, consider the exponential function

By definition, the natural logarithm of y will be x because that is the power to which e is taken to equal x Thus we can write

If we only know the value of the natural logarithm ln y and wish to find y then, by substituting

(2) into (1), it must be true that

Determination of continuous growth rates using natural logarithms

To understand how natural logarithms can help determine rates of continuous growth, considerthe following example

Example 14.9

The consumption of natural mineral resource M has risen from 38 million tonnes (per annum)

to 68.4 million tonnes over the last 12 years If it is assumed that growth in consumption hasbeen continuous, what is the annual rate of growth?

Solution

If growth is continuous then the final consumption level of M will be determined by theexponential function:

This time the known values are: the final value M = 68.4, the initial consumption value

M0= 38, and t = 12, with the rate of growth r being the unknown value that we are trying

and so consumption has risen at an annual rate of 4.9%

A general formula for finding a continuous rate of growth when y, A and t are all known

can be derived from the final sum formula Given

where C0is the catch in the initial time period.

Substituting the known values into this function gives

Therefore the rate of decline is 1.8%

Rates of growth and decay can also be determined over time periods of less than a year byemploying the same method

Therefore the annual growth rate is 6.84%.

Comparison of discrete and continuous growth

A direct comparison of the continuous growth rate r and the discrete growth rate i that would accumulate the same final sum F over 1 year for a given initial sum A can be found using

natural logarithms, as follows:

Continuous growth final sum F = Ae r

Discrete growth final sum F = A(1 + i)

Test Yourself, Exercise 14.3

1 In an advanced industrial economy, population is observed to have grown at asteady rate from 50 to 55 million over the last 20 years What is the annual rate ofgrowth?

2 If the average quantity of petrol used per week by a typical private motorist hasincreased from 32.1 litres to 48.4 litres over the last 20 years, what has been theaverage annual growth rate in petrol consumption assuming that this increase inpetrol consumption has been continuous? If, over the same time period, petrolconsumption for a typical private car has fallen from 8.75 litres per 100 km to 6.56litres per 100 km, what has been the average annual growth rate in the distancecovered each week by a typical motorist?

3 World reserves of mineral M are observed to have declined from 830 million tonnes

to 675 million tonnes over the last 25 years Assuming this decline to have beencontinuous, calculate the annual rate of decline and then predict what reserves will

be left in 10 years’ time

4 An economy’s GNP grows frome5,682 million to e5,727 million during the first

quarter of a new government’s term of office If this growth rate persisted through

its entire term of office of 4 years, what would GNP be at the time of the nextelection?

5 If the number of a protected species of animal in a reserve increased continuallyfrom 600 in 1992 to 1,450 in 2002, what was the annual growth rate?

6 Over a 12-month period what continuous growth rate is equivalent to a discretegrowth rate of 6%?

7 What discrete annual growth rate is equivalent to a continuous growth rate of 6%persisting over 12 months?

8 What interest rate would you prefer to be used to add interest to your savings: 8%applied on a continuous basis or 9% applied once a year?

14.5Differentiation of logarithmic functions

We have already used the rule that if y= et then dy/dt = et This result can be derived if weaccept as given the rule for differentiation of the natural logarithm function This rule saysthat if



1

x = ln x

This was the exceptional case in integration not dealt with inChapter 12

Returning to the exponential function, we write this with the two sides of the equality sideswapped around Thus

Substituting (1) into (2) gives

which is the result we wished to prove

14.6 Continuous time and differential equations

We have already seen how continuous growth rates can be determined and how continuousgrowth affects the final sum accumulated, but to analyse certain economic models wherecontinuous dynamic adjustment occurs we also need to understand what differential equationsare and how they can be solved

Differential equations contain the derivative of an unknown function For example

dy/dt but not higher order derivatives such as d2y/ dt2 Linear means that a differential

equa-tion does not contain a product such as y (dy/dt) More advanced mathematical economics

texts will cover the analysis of higher-order and non-linear differential equations

As well as containing the first-order derivative, a first-order differential equation will

usually also contain the unknown function (y) itself Thus a first-order differential equation

may contain:

• a constant (although this may be zero)

• the unknown function y

• the first-order derivative dy/dt

At first sight you might think integration would be the way to find the unknown function

However, as a differential equation will include terms in y rather than t the solution is not

quite so straightforward For example, if we had started with a basic derivative such as