02-symbols, formulas and rules

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The UNIVERSITY of GREENWICH

MATHEMATICS FOR ECONOMISTS 1999-2000

MODULE 2: SYMBOLS, FORMULAS AND RULES

Case Study:The consumption function

In book III, Chapter I of the General Theory of Employment, Interest and Money, The English

economist John Maynard Keynes argues that

"The fundamental psychological law, upon which we are entitled to depend with great confidence

both a priori from our knowledge of human nature and from the detailed facts of experience, is that

men are disposed, as a rule and on the average, to increase their consumption as their income

increases, but not by as much as the increase in their income"

A great deal of macroeconomics is devoted to translating this into mathematical terms, and

deducing from it the rules which govern a market economy. This is what we shall study in this unit. In

summary there are two questions:

How do you turn Keynes's statement into a mathematical formula?

Can you use your formula to make predictions about the economy?

PRELIMINARY: EXPRESSING FACTS AS FORMULAS

Before returning to the consumption function we will deal with a simpler problem which also helps

us understand what Keynes was trying to express. This problem is as follows:

How can we use mathematics to express the basic facts of the economy?

If you buy, rent, borrow or copy the national accounts of the British economy otherwise known as

the Blue Book you will find, in the second table, the figures given in table 1

In fact you do not need the last row. You can always calculate Gross Domestic Product by adding and

subtracting the other figures in the table. In the computing section of this course you will learn how

to put these figures into a spreadsheetto do this calculation automatically.

Suppose for now that you do not have a mechanical computer, but you are going to hire a drudge

a human computerto do the calculation. How would you relay your instructions? You could just

repeat the rule in English. Or you could do it in any other language of your choosing, provided the

computer understood.


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Module 2: Symbols, Formulas and Rules Alan Freeman

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1968 1969 1970 1971 1972

Consumers' Expenditure 27751 29466 32114 36010 40750

General Government final consumption 7681 8018 9038 10305 11751

Gross Domestic fixed capital formation 8506 8832 9736 10894 11940

Value of physical increase in stocks and work in progress 452 537 382 114 25

Exports of goods and services 8980 10087 11510 12918 13621

Imports of goods and services 9380 9930 11103 12161 13740

Gross Domestic Product 43990 47010 51677 58080 64347

Table 1: extract from the National Income Accounts of the United Kingdom

Or ... you could repeat the rule in the language of mathematics. Instead of referring to Consumers'

Expenditure, you could say C. Instead of General Government Final Consumption, you could say G

and so on. Then you could explain what must be done to C, Gand all the rest in order to calculate

GDP by simply writing them as a sum:

GDP = C+ G+ ... etc.

Think through the process by which you might do this. Four steps are involved

You have to identify the quantities involved

You have to choose names (symbols) for them

You have to write down the calculation in ordinary English

Your translate the English expression into mathematics

The same process is followed, more or less, in writing out any and every theory in mathematical

form. Now we look at each of these steps in detail

STEP 1: IDENTIFY THE QUANTITIES

If an economic problem can be treated mathematically, this can be done in one of two ways:

Using algebra, by means of equations

Using geometry, by means of graphs

In either case, the first step is to identify the quantities involved. This consists of everything which

can have numbers attached to it. For the national accounts this is easy: they are in a table with the

names down the left hand column, namely.

Consumers' Expenditure

General Government final consumption


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Gross Domestic fixed capital formation

Value of physical increase in stocks and work in progress

Exports of goods and services

Imports of goods and services

Gross Domestic Product

For Keynes's statement matters are a bit more difficult. He refers to:

fundamental psychological law

great confidence

knowledge

human nature

the facts of experience

men

consumption

income

increase in their income

Only three of these can be quantified: turned into numbers. They have been marked out in bold. The

mathematics used by economists does not try and attach symbols to anything else

STEP 2: DECIDE WHAT SYMBOLS TO USE

The second step in translating any economic idea into mathematical language is just this: deciding on

the names which you are going to give to the quantities involved.

In economics, some quantities have come, by convention, to be represented by certain letters, and it

is a useful policy to stick with this. Some common abbreviations are given below. You don't have to

use these letters. You could choose, say, Nor NIto represent National Income. But if you do so, your

reader might be confused. Like Humpty Dumpty, you can use a word to mean exactly what you want

it to mean. Unlike Humpty Dumpty, you have to tell your reader what it means. You should tell her

or him how you intend to use your symbols with a statement like

Let NIstand for National Income.

Or Using NIto represent National Income,...

or (after the first equation where you use it)

...where NIstands for National Income.


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COMMON ECONOMIC SYMBOLS

C private consumption (what households spend each year)

G government spending (what the government spends each year)

I Investment (what firms invest each year, usually net

after depreciation)

Y National Income (what the whole economy spends each year)

M Money or, sometimes, Imports

X Exports

i,r The rate of interest

T Tax income during the year

P Price

Q Quantity

S Supply

D Demand

Scripts and Subscripts

You would have to exercise some care. You might want to use the symbol Ito represent Gross

Investment. This is perfectly reasonable but you should be aware that there are several different

ways of defining investment, for example:

Gross Investment (before allowing for depreciation)

Net Investment (after allowing for depreciation)

Since the symbol I is more usually used for net investment, you can either

Use Ito represent gross investment, but include a statement saying what you have done,

or:

Use a new symbol for gross investment

What symbol can you use for Gross investment? You might use a letter chosen at random J,

perhaps. Since there are only twenty-six letters in the alphabet, you might resort to greek letters and

call it , or even use a gothic letter. One reason for the bizarre appearance of some mathematics is

that mathematicians run out of letters and use weird scripts. It also makes the work look learned.

An alternative, preferred by economists, is to use two letters and make one a subscript. So you could

call it IG, for Gross Investment. The advantage of this is that it is clearer: you can see that it is

investment, but a special kind of investment

the gross kind.


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Likewise we need a symbol for Gross Domestic Product, since the conventional symbol Y,reserved

for National Income, is adjusted from the Gross Domestic Product by subtracting indirect taxes,

adding income from abroad, and subtracting capital consumption. You could say, for example

Let YGbe Gross Domestic Product

!:

prefixes and their uses

Since the increase in stocks doesn't have a conventional symbol, you would have to invent a symbol

for it. Again, you could just pick any letter. However, mathematics provides a useful, if idiosyncratic

special notation for anything involving an increaseor change. This consists using the greek letter

as prefixputting it in front of a normal letter.

If we use the letter S, therefore, to mean stocks, then the expression

S

means 'the change in S'. In this case, it simply means the difference between what it was last year,

and what it is this year.

This special use of symbols is another reason why mathematics texts appear daunting. But in this

case there is a good reason, which is that it makes for very brief notation. A great deal of

mathematics is no more than careful notation, designed to free the mind from the clutter of great

strings of words.

To read this expression out loud, you can either say 'Delta S' or 'the change in S'. You will meet the

symbol in two contexts

Representing change over time in this case the change in the value of stocks from one

year to the next.

Marginalist economics use it to represent a marginal(small) change in some quantity

You will encounter the symbol most frequently in statistics and econometrics

Two special symbols used in mathematics and their translation

Means the change in

means the sum of

STEP 3: WRITE OUT THE INSTRUCTIONS IN ENGLISH

The rules for calculating GDP is fairly simple and laid down by the people who wrote the Blue Book.

It is as follows:

Rule for working out Gross Domestic Product:


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Add up Consumers' Expenditure, General Government final consumption, Gross Domestic

fixed capital formation, Value of increase in stocks, and Exports, and then take away

Imports.

For Keynes's formula matters are a bit more complex; we shall come back to it.

STEP 4: TRANSLATE THE RULE INTO MATHEMATICS

We can begin translating your instructions into mathematics, as follows:

To calculate YG, add up C, G, IG, S and X, and then take away M.

But why stop at replacing the quantities? Mathematics provides a shorthand for every word in the

sentence and we can rewrite it as

YG= C + G + IG+ S + X M

Such a shorthand notation is known as a formulaor, since it has an equals sign in it, an equation. We

can now calculate GDP from the other economic statistics in the Blue Book.

Theory and factan important distinction

A far as a mathematician is concerned, one formula is no different from another. As an economist,

however, you should learn to distinguish two types of formula from very different sources, namely:

Formulae that are true by definitionso-called accounting identities

Formulae that are derived from a theory.

The rule we gave above is a National Accounting Identity. It is true because the statisticians cook thebooks to make sure the figures add up. To look at it another way, the symbol YG isjust another way

of writingC + G + IG+ S + X M.

Aside: translating algebra to english

Your problem will often be to understand a formula that has been given to you for example

C= 100 + 0.8Y

The rule is the reverse: wherever you see a symbol, replace it with the corresponding english

language description. Wherever you see a number, leave it alone. So the rule means

Consumption is equal to 100 added to0.8times national income

The consumption function is a different kettle of fish altogether. An accounting identity mustbe true

by definition. The consumption function can only hold if the theory behind it is true. It has to be

checked against experience.


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FORMALISING KEYNES'S THEORY

Keynes was very cautious. He did not say that consumption bore a precise relation to income; he

only said that it would increase by less than it.

Because of this, we cannot yet go any further forward in writing down a mathematical formula for

the relation between consumption and income. In order to do so, we need a precise theory of

consumption. If it is a useful theory, it should provide us with a measureable relationbetween Cand

Y.

You can study this in more detail in the macroeconomics course or in any textbook which discusses

the consumption function. Some references are given at the end of these notes. During the rest of

this course we will use a very simplified version of the consumption function which many economists

believe to be a reasonable approximation to what happens in the real world.

The theory is as follows: for each 1 rise in national income, people will save a certain proportion

for example, they might save 1/5 of it, or 20p. The rest

80p or 4/5 of it, they will consume. In

other words,

whatever the increase in income, consumption will increase by 4/5 as much

How can we translate this into mathematics?

Steps 1 and 2: choose quantities and name them

The concepts which can be quantified appear to be as follows:

income

consumption

increase

Reading Keynes carefully, you can see that he has nothing to say about the actual sizeof either

consumption or Income. In fact he talks about the increasein income and the increase in

consumption.

'Increase' or 'change' always translates as the greek letter . For consumption and income we can

use the conventional symbols Cand Y. To summarise

C means consumption

Y means income

C means the increase in consumption

Y means the increase in income

We shall shortly find that this naming process is not quite complete; there is a 'hidden' quantity in

Keynes's statement. We shall come back to this. This is very typical of the creative process of writing


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a mathematical formula. Steps 1 to four are repeatedly followed, not just once but several times

until the final product is ready. For now, we go to steps 3 and 4.

Steps 3 and 4: write down a formula and translate it into mathematics

The statement marked above can be slightly rewritten to make our substitution easier, as follows:

The increase in consumption will be 4/5 as much as the increase in income

But the words 'as much as' obviously just mean 'times'. So the formula reads

C= 4/5 Y

or, in decimals

C= 0.8 Y

A new problem: the hidden quantities

We can make the statement a bit more general. To illustrate what we were saying we 'supposed'

that consumption increased by 4/5 of the increase in income. In fact it might not be 4/5 but 3/5, 0.9,

0.95, or any figure less than 1. The exact amount has to be discovered by investigation, so it is not at

present known.

Mathematics has a simple procedure for dealing with things which are not known: it labels them

with a symbolusuallyx, the unknown.

Most people think that science explains things that we don't understand in terms of things that we

do. In fact it explains things that we do understand in terms of things that we don't

Bertrand Russell

Using symbols for unknown quantities is yet another reason that mathematics appears daunting. The

average mathematics text read like an astrologists' love-letter. It is littered with arcane symbols for

unattainable things.

But in fact naming unknown quantities imposes a rather important discipline: it brings out your

hidden assumptions. In this case it makes it absolutely clear that there is an economic assumption

involved in this theory, which is that Cstands in some fixed, constant ratio to Y. Keynesian

economists are so convinced of the correctness of this assumption that they apply a name to thisratio: it is called the Marginal Propensity to Consume.

Of course, if the economists' assumptions are false, then the justification for attaching a name to

their unknowns is much weaker. If there is no relation between C and Y, there is no basis for

talking of the marginal propensity to consume as if really existed. One advantage of mathematical

notation is that, if it is done properly, it turns the spotlight on your pet theories. For now, we assume

the Marginal Propensity to Consume is real, and use mfor it. Our final version of the formula is now

ready:

C= m Y (1)


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Using formulae

Y = C + G + I + X M

is a rule for calculating. Given C,G,I, X and M, it tells us Y. For example, if

C =400 G=100 I =20 X=100 M=20

then Yis 400+100+20+30+10020 = 630. The process is called substitution.

The substitution rule

replace each letter by the number it stands for.

Alternatively, and more generally

replace each letter which appears on the left-hand side of an equals sign, by whatever you find on

the right-hand side of the equals sign

This it has much more uses. Suppose we are nottold Cbut we know all the other figures:

C =? G=100 I =50 X=100 M=20

and Y= 700. What is C? Evidently, it is a number which makes the following equation true

700= C+ 100 + 50 + 100 + 20

We solve this by an algebraic operationwhich is one of the rules of algebra: we take all the numbers

across from the left to the right, and change their sign. This gives

700

100

20

30

100 + 20 = C

or, adding up, 470 = C

Another rule says we can reverse the two sides of the equation to give C= 470. But we can go one

step further. We can re-arrange the formula to give instructions for calculating Cinstead of a

calculating Y. We can apply the same rule of algebra beforesubstituting numbers to give us

Y GIX + M= C

and hence C= Y GIX + M (2)

Check this out: cover up the row in the table on page 1 and use the formula to calculate Consumers'Expenditure in 1974. You should calculate I as IG+S (gross investment plus change in stocks and work

in progress)

Even better: suppose we are given the value of everything except C and M.


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Y=700 C =?

G=100

I =50

X=100

M=?

We can now create a formula for calculating Cwhen we get a value of M. Substitute into (2) all thenumbers we are given, but leave the letters Mand Calone. This gives

C= 70010050100 + M

or, adding up, C= 650 + M

Such a formula is called aformula for C in terms of M. It is a rule for calculating Cfrom M,provided

that all the other quantities are held fixed.

This is the raw stuff of algebra. We are using a rule, which describes knowledge we already have, to

deduce other rules and give new knowledge we do not have. But it arose from a simple idea: tospeed up calculations by re-arranging the terms in our original formula.

Trust me, I'm a mathematicianmaking Keynes' formula useful

The formula (1) above can be used in the same way as the national accounting identity. Given values

for mand Yit will tell us what Cis. Thus

If mis 0.8 and Yis 10, Cis 0.8 10 = 8

If mis 0.7 and Yis 20, Cis 0.7 20 = 14

and so on.

But it doesn't tell us what Cis only the change in C, that is C. In order to find out Cafter it has

changed, you would have to know what it was before it changed. If you are told a child is one year

older, you cannot say how old it is unless you know how old it was a year ago.

Later on we will discuss how to connect up Cand C. For now, suspend judgement. Accept that if a

formula like (1) is true, we can deduce another formula connecting Cand Y:

C= a+ mY

where ais a new unknown called 'autonomous consumption'. In theory, it is equal to the amount

which is consumed when nobody has any income at all. You don't have to take this entirely on trust:

you can check it out by conducting experiments with numbersto verify it.

EXPERIMENTING WITH NUMBERS:LEARNING HOW A FORMULA WORKS

When you have become familiar with expressions like C= a+ mYyou will develop a 'feel' for how

they work in exactly the same way that you develop a feel for steering a car. It will cease to be a

collection of meaningless symbols. One of the aims of the early part of this course is to give you this

familiarity through use.


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How do you use a formula? In exactly the way it was intended: to calculate things. This formula is

intended to tell you the relation between Consumption and Income. That means: if I tell you what

Income is, you can tell me what Consumption is using the formula.

But as the formula stands there are too many unknowns to complete the calculation. If you want to

calculate Cit is not enough to know Y: you have to know aand mas well.

The standard way of dealing with this, which you will study in Statistics, is to find aand mthrough

investigation. You don't have to do this in order to find out how the formula works. There is nothing

to stop you inventingvalues for aand mand simply seeing what happens. You could say, for the sake

of argument, suppose

a=100

m= 0.9

The formula now becomes C= 100 + 0.9Y

mcannot realistically be greater than 1 or less than 0. Why not?

In one of the exercises for this unit you are asked to study whether, in fact, the relation

C= m Y

might actually hold. For now, we shall conduct a different experiment with numbers: we will use a

simple formula, together with a simplified version of the national accounting identity, in order tosolve the problem defined at the start of this unit, namely:

Can we use the formula to make predictions about the economy?

A SIMPLE MODEL OF INCOME DETERMINATION

We now begin work on a simple experiment with numbers, using two formulae

Formula 1: a simplified national accounting identity

Y = C + I + G

Formula 2: a simple Keynesian consumption function

C = 100 + 0.7Y

As a further assumption, suppose that we already know the level of investment and government

spending, and they are given by


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I = 10

G = 20

Can we use these formulae to determine Cand Y?

The last link in the chain: the equilibrium theory of income adjustment

Earlier in this unit it was explained that Formula 1 and formula 2 are different. Formula 1, an

'identity', is true by definition. It can be thought of as a fact. Formula 2 is a theory.

There appears to be a possible problem. What if the theory contradicts the facts? What if income

and consumption don't satisfy both formulae? A great deal of discussion has gone into this problem,

which is at the heart of what modern macroeconomics has to say.

There are a variety of approaches which we won't go into here. One approach is to say that either

the consumption function, or the accounting identity, does not refer to actual spending but to

spending 'intentions'. From this arises the distinction between so-called ex-postand ex-antefigures;

roughly speaking ex-anterefers to what you hope for and ex-anterefers to what you get. The

economy is then supposed to adjust until the two are in harmony.

Another approach, which we shall adopt here, is to regard formula 1 as a statement about what the

economy spends(aggregate demand or expenditure). The Yin this formula therefore refers to the

total outgoings of private consumers, government and investors. The Yin formula 2 has a slightly

different meaning. It refers to what the economy receives(national income). Strictly speaking, since

the two versions of Y do not refer to the same thing, we should use different letters for them. The

fullblown version of the model should read

YE= C + I + G

C= 100 + 0.7Y

I = 10

G= 20

where YEmeans 'expenditure on the national income'. According to this theory, supposed to adjust

until expenditure matches income; this means we need an extra formula

Y = YE (3)

Very often, however, writers do without this degree of precision and assume that since Y = YEthey

can just substitute Yimmediately wherever YEappears and so use Yindiscriminately throughout.

There is no harm in it but it sometimes makes it harder to understand what the mechanism of

adjustment in the model actually is supposed to be. In fact, as we shall discuss in more detail in unit

4, equation (3) actually represents the mechanism of adjustment.


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A theory of income adjustment

What does 'harmony' mean? In precise mathematical terms, Yand Cmust have the same value in

both formulae.In short, both formulae must be simultaneously true. We now study the mathematics

that tells you what Cand Ymust be.

THE RULES OF ARITHMETIC

To solve the problem we are going to apply the rules of algebra to the formula to find out what Y is.

The best way to understand the rules of algebra is to realise that they are nothing more than the

rules of arithmetic applied to symbols.

First we shall review some of the basic rules of arithmetic. If you are unsure of any problem in

algebra, it is always a good idea to work it through with ordinary numbers first to get the feel of it.

There are about five rules we will use again and again. Get used to them. If you have any doubt

about whether you are happy with using them, work at them now until you can use them. If

necessary use the special revision class set aside for this purpose. The rules deal with:

minus signs

expanding brackets

multiplying fractions

adding fractions

dividing through by common factors

THE RULES FOR MINUS SIGNS

In arithmetic and in algebra, combining two minusses makes a plus, combining plus with a minus

makes a minus.

(2) is +2

(+2) is2

+(2) is -2

+(+2) is +2

2 2 is +4

2 2 is4

2 2 is4

2 2 is +4

2 2 is +1

2 2 is1

2 2 is1

2 2 is +1

EXPANDING BRACKETS

Whenever you see an expression like

2 (3 + 4)

It can be replaced by 2 3 + 2 4


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You get this by applying the multiplier to each item inside the brackets or 'taking it inside the

brackets'. This is a very fundamental law of arithmetic. It works with minus signs as well:

2 ( 45)

= 2 42 5

And no matter how many terms are inside the brackets

2 ( 3 + 45 + 6)

23 + 2425 + 26

It also works when the multiplier is on the right of the brackets:

(3 + 4 ) 2 = (32 + 42)

WHEN TO BE CAREFUL WITH THE BRACKETS RULE

The rule doesn't work:

With or / signs inside the brackets

2 (3 4) is not 23 24

With + or signs outside the brackets

2(3 / 4) is not 23 / 24

With / on the left:

2 / (3 + 4) is not 2/3 + 2/4

Does it work with / on the right? i.e. is true that:( 3 + 4) / 2 equal to 3/2 + 4/2?

These rules of arithmetic are described in set theory more precisely:

Multiplication is 'Distributive with respect to addition and subtraction'

a (b + c) = ab + ac

Multiplication and addition are 'commutative'

a b = b a and a + b = b + a

Is division commutative? Is it distributive with respect to addition?


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SPLITTING UP FRACTIONS USING THE BRACKETS RULE

With fractions the brackets rule can apply where you don't realise it

4 8

2

4 8

2

4 8 2

is really that is( )

( )

so it can be rewritten as4

2

8

2

MORE COMPLEX EXPRESSIONS

When multiplication or division signs appear inside brackets, each little group of symbols,

connected together by these signs, is treated like a single number. These are called terms:

in 2 ( 34 +4/5)

3 4 is a term and4/5is a term

So multiply each term by 2 2 3 4 + 2 4/5

When you have two groups of brackets to multiply, just use the rule twice:

(2 + 3)(4 + 5)

2(4 + 5) + 3 (4 + 5)

2 4 + 25 + 3 4 + 35

LONG MULTIPLICATION REVISITED

We can take a second look at long multiplication using the bracket rule. There is more to this than

meets the eye: we shall come back to it when we deal with power functions and logarithms.

Remember that a number like 132 is really

1100 + 310 + 21

Long multiplication is really done by expanding brackets:

132 21 is the same as

(1100 + 310 + 21)

(210 + 11)

Which, by the bracket rule, is

210(1100 + 310 + 21)

+ 11(1100 + 310 + 21)


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21000 + 6100 + 410

+ 1100 + 310 + 21

21000 + 7100 + 710 + 21

The arabic number system lets us line everything up in columns and miss out the 100 and 10multipliers, because we know from the position of a numeral whether it represents units, tens,

hundreds or whatever. So we can write this as

2 6 4 0

+0 1 3 2

=2 7 7 2

Our number system works in multiples of ten. Computers work in multiples of 2. This is called the

binary system: we can write any number in binary as a string of ones and zeroes. The number 132 ,for example is

10000100

which simply means

127+ 026+ 025+ 024+ 023+ 122 + 021 + 01

FRACTIONS

Division can be written in several ways:

1/3 or 1 3 or1

3

The basic rules of fractions deal with

multiplying fractions

dividing fractions

adding and subtracting fractions

cancelling common factors

MULTIPLYING FRACTIONS

The rule is: multiply the top parts, and multiply the bottom parts:

2

3

4

5

2 4

3 5


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DIVIDING FRACTIONS

The rule is: turn the second fraction upside down and change the division sign to a multiplication

sign

2

3

4

5

2

3

5

4

2 5

3 4

In economics you often have to divide by a decimal, for example

100

0 5. Multiply both the top and the bottom by 10 to remove the decimal, then do the division

100

0 5

100 10

0 5 10

1000

5200

. .

If there are two figures after the decimal point, multiply by 100 ... and so on

ADDING FRACTIONS

Adding and subtracting fractions are dealt with in the next section on algebra, because they are

easier to understand with symbols than with numbers.

HOW THE RULES OF ARITHMETIC CARRY OVER TO ALGEBRA

Every rule of arithmetic has its counterpart in algebra. The same rules work, if you put a letter in

place of any number.

The minus rule: (x) = +x

+(x) =x

(+x) =x

x (y) = +xy

x/y = +x/y

The brackets rule: 2(x+y) = 2x+ 2y

(x+ y)/2 =x/2 + y/2

More complex brackets (2x+ 3) (y4)

= 2x(y4) + 3(y4)

= 2xy4x+ 3y12

ADDING AND SUBTRACTING FRACTIONS

This is the most complex operation of all, and a lot easier to write using algebra than describe in

english:


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a

b

c

d

ad bc

bd

example2

3

5

7

2 7 5 3

3 7

This may look a lot simpler than the rules often taught in school, which involve Least Common

Multiples, Common Factors, and so on. It is. Adding fractions is actually very simple, but it can give

you quite large multiplications to carry out. All the business with Least Common Multiples is simply

designed to cut out the work of multiplication. Unfortunately it makes the rules more complicated.

If you are happy with the method you were taught at school, use it. If not, the method above is

the simplest to remember, and it will always work

You often have to add a number to a fraction, like

100 + 300/7

A whole number is like a fraction where the denominator (the number underneath) is 1

100 300

7

100

1

300

7

100 7 300 1

1 7

= 1000/7

CANCELLING COMMON FACTORS

You can always cut down some of the work of adding and multiplying fractions if you can cancel out

common factors. This is also easier in algebra than it is in arithmetic The rule says that if the same

symbol or number will divide into both the top and the bottom of a fraction, then you can divide

through by it top and bottom and so eliminated it.

2a/3a is the same as 2/3 (divide top and bottom by a)

15/10 is the same as 3/2 (divide top and bottom by 5)

This rule is often more important in algebra because it can be used to simplify complex expressions.

Any common factor will do, even one in brackets.

for example4 2 3

5 2 3

4

5

( )

( )

x

x

because it can be divided top and bottom by 2x+3

Caution: Cancelling only works when things are multiplied together.

A common mistake is

100

0 5100

0 5

G G

. .

Correct is to divide 50 into both terms on top:

100

0 5 0 520

0 520 2

. . .

G GG

If you make this mistake look at the section entitled 'splitting up fractions using the brackets rule' above and review

the computer tutorial on the distributive rule


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COLLECTING TERMS:THE BRACKETS RULE IN REVERSE

Often in algebra we get to a point where we cannot do any more arithmetic, but the expression

could still be simplified. The reason is fairly subtle: it is because we cannot arithmetically combine

two different symbols without knowing their value. This is different from arithmetic where we can

always reduce any expression to a single number. Thus

3 ( 4 + 5)

3 4 + 3 5

12 + 15

27

but with x and y in place of 4 and 5:

3 (x+ y)

3x+ 4y

and that is as far as it goes.

However, sometimes the expression can be simplified further if there is more than one term in the

same unknown. We can addxto 2xto get 3x, or take 4xfrom 3xto getx.

The usual practice is to group together all terms which have the same unknown by reordering the

terms in the expression. Look at this problem:

Simplify 3x+ 4y2x+ 4

Begin by grouping the terms with an xin them

(3x2x) + 4y

There is then a rule which says we can reverse 'expanding the brackets' by taking the unknown

outside the brackets:

(32) x+ 4y

Next take 2 from 3 to give 1

1 x + y

Final answer x + y

SUBSTITUTION:TAILOR-MADE FOR MACROECONOMICS

You can now begin work on the case study. You are told that


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Y = C + I + G

C= 100 + 0.7Y

I= 10

G= 20

Systems of several equations like this are typical of neo-classical macroeconomics and the theory of

income determination. The method best suited to solve them is called substitution.

At school you will probably have prioritised another method for solving simultaneous equation

systems, called elimination. You will revisit this later in the course. It is a more general method;

however substitution is much better suited to the equations that are thrown up by neoclassical

macroeconomics.

The basic rule of substitution is this: whenever you find a single symbol on the left of an equation

(for example G= 20), then wherever you find this symbol it can be replaced (substituted for) by what

you find on the right. So wherever you find Gyou can substitute 20.

You can immediately make two simplifications. You are given a value for I, and for G. Therefore you

can write these two values wherever you see I or G in the formulae above. Each time you do this you

eliminate one of the symbols. When you solve an equation in algebra, the first thing you do is start

eliminating symbols, untilideallyyou have only one symbol, and your expression tells you its

value.

First substitute for I. This gives

Y = C + 10 + G

C= 100 + 0.7Y

G = 20

Next substitute for G. This gives

Y= C+ 10 + 20

C = 100 + 0.7Y

Finally you can carry out a bit of arithmetic to get

Y = C+ 30

C= 100 + 0.7 Y

Now what do you do? The trick is to remember that substitution also applies when Cstands for an

expression and not just a number.

The second equation tells you that Cis 100 + 0.7Y. This can be substituted for Cin the first equation

to give


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Y= 100 + 0.7Y + 30

THE RULES FOR SOLVING EQUATIONS

You now have an equation in Y. To solve it you have to use rules which are unique to algebra. They

have no equivalent in arithmetic. The aim of nearly all algebraic manipulation is

First, to eliminate all unknown symbols except one, which should figure in just one

equation (like the one above)

Then, to fiddle around with the equation until the unknown is on one side of the = sign,

and everything else is on the other. The result is a formula with which we can calculate

the last remaining unknown. In short, you aim to get something like:

Y= (an expression with only numbers in it)

The procedure is mechanical. Like the rules for arithmetic, you should learn it because you will use it

again and again. If you have trouble with it you must practice until you can do it, come to the

revision class and/or work through the computer tutorials in basic arithmetic.

If you are learning to drive and you can't steer, you won't be able to drive. Manipulating linear

equations (equations like these) is the same as steering. It takes practice; everyone can learn it, but

you have to learn it or nothing else will make sense.

THE 'TAKING ACROSS'RULES

There are two new rules to learn. These rules only apply to equations. They are:

'taking across' terms which are added or subtracted

'taking across' multipliers or divisors.

Taking across terms that are added or subtracted.

You can take any term on either side of an equation which is being added or subtracted, and move it

to the other side, provided you change its sign.

So 2x+ 3 = 4x

can be turned into 2x+ 34 =x

or 2x+ 3 = 4x

can be turned into 2x+x= 4

This rule takes you one step further with the case study. You can start collecting all the terms that

involve Yon the left, and all the numbers on the right. From

Y= 100 + 0.7Y+ 30

add the numbers up Y= 130 + 0.7Y


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take terms in Yto the left Y0.7Y= 130

simplify (see box) 0.3Y= 130

Tip on simplifying: the move from Y0.7Y to 0.3Y is often puzzling. It involves a typical trick which

recurs again and again in this type of problem. You have to remember that Y is the same as 1 Y. The

expression really reads1 Y 0.7 Y

You treat this exactly as if it were, say, 3Y2Y , which is (3 2)Y =Y. It becomes

(1 0.7) Y = 0.3Y

If you have trouble with any of these steps, look back to Collecting terms - the brackets rule in

reverse, read the box on simplifying just above, and work through the computer tutorial on solving

equations.

Final step: Taking across multipliers or divisors

The last rule is similar to taking across added or subtracted signs. It is sometimes called 'multiplying

through' or 'dividing through'. It says that we can divide both sides of the equation by the samething, or multiply them both by the same thing. This is used to reduce the left hand side expression

0.3Yto a plain Yand solve the equation:

divide by 0.3 Y= 130/0.3

Y = 1300/3 = 4331/3

The very last step is now to find the value of C. You get this by substituting yet again. This time we

take the value of Yyou just worked out and putting it back into the formula for C which you already

have, namely the consumption function

C= 100 + 0.7 Y

substitute Y = 13001/3 C = 100 + 0.7 1300/3

=100 + 7 130/3

= 100 + 910/3

= 100 + 3031/3

= 403

1

/3

Fractions, decimals and accuracy

You may have blanched at the fractions in the last few lines. You can use a calculator if you prefer.

You do lose something, however. A fraction is exact, but a decimal may not be. Your calculator will

represent 1300/3 as 433.3333333, which you probably keep to two decimal places as 433.33. Now,

if you substitute this back into the consumption function you lose still more accuracy. Make sure,

therefore, that you keep enough decimal places so that after a sequence of substitutions, you don't

start to go seriously adrift. A rule of thumb is: while you are still doing intermediate calculations,

keep two extra places of decimals.


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A better procedure is to use fractions; it is tedious but it is part of practicing the basic arithmetic

operations which should become second nature to you.

WAYS TO SKIN A CAT

There is no rule that says you have to begin by substituting for Y. You could have got the sameresults another way, by substituting for Yin the consumption function (after substituting for Iand G

to get Y= C+ 30). This gives

C = 100 + 0.7(C+ 30)

Now you can begin to see why it is so important to be able to expand brackets. When substituting a

whole expression, like C+ 30, you should always put brackets round it: if we had written

C= 100 + 0.7 C+ 30

The result would be wrong. The whole expression which stands where Yused to stand must be

multiplied by 0.7 and not just part of it. Expanding brackets now gives

C= 100 + 0.7C+ 21

= 121 + 0.7C

Now collect terms in C: C0.7C = 121

Simplify (10.7)C= 121

0.3C= 121

Divide through by 0.3 C= 121/0.3

= 4031/3

It is a good idea, if there is more than one way to solve an equation, to do it two different ways and

make sure you get the same answer.

LAST OF ALL:CHECK THE WORK

In school, you are often made to feel that you have to prove you can do mathematics by producing

answers quickly, in your head, and generally demonstrating your prowess.

In any real situation where mathematics is used this is a completely useless skill. Far more important

is getting the right answer regularly and without fail. An engineer who works out how to build a

bridge in half the time is not much use if the bridge falls down.

There are many reasons for learning standardised procedures in calculation. boring though it may

seem. It helps to learn because of the practice it gives; it makes for collective work, because other

people can follow what you were doing; it makes for good revision, because youcan follow what you


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were doing, six months later. Most of all, however, standardised procedures give a better guarantee

of getting the answer. And the last part of a standardised procedure is to check your answer.

In this course, if you fail to check your answer in a situation where it is possible to do so, you will lose

marks. You will also getmarks for checking your work, even if the answer was wrong provided

your check has shown that it is wrong.

You cannot always check your answers, but when solving equations you can: by substituting the

answers you got back in the original equation.

In this case the solutions are

C= 4031/3

Y = 4431/3

and these must be substituted, along with I= 10, G = 20 in the two main equations

C= 100 + 0.7 Y (4)

and Y= C+ I+ G (5)

Substituting in (4) 4031/3= 100 + 0.7 4431/3

= 100 + 0.7 1300/3

= 100 + 910/3

= 100 + 3031/3

= 4031/3

Substituting in (5) 4431/3= 4031/3+ 20 + 10

USING FORMULAS TO PRODUCE MORE FORMULAS

What happens if government expenditure changes? Our last calculation is no longer valid. However,

you can of course repeat it with the new value for G, say

G= 15

The same rather laborious calculation will finally give

Y= 4111/3

Now suppose you are asked to repeat it again with G=10. By now it is getting a trifle tedious to

repeat the same calculation each and every time. But there is a quicker way. You can use the original

formulae to produce a formula for Ybeforewe know what Gis. Then you can use this formula to

calculate Yquickly from Gand I by substituting.


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The procedure is to 'solve' the equation in the same way as when you know what Gand Iare:

C= 100 + 0.7Y

Y = C+ G+ 10

Y= 100 + 0.7Y+ G+ 10

Y= 100 + 0.7Y+ G+ 10

Y0.7Y= 100 + G+ 10

0.3Y= 100 + G+ 10 = 110 + G

Y= (110 + G)/0.3

Y= 1100/3 +G/0.3 = 3662/3+ G/0.3

This is a new formula which tells us how to calculate Y from G . We can use it to solve our original

question by substituting G= 15 to give

Y = 3662/3+ 45= 4112/3

But it is much more useful than this: we can now use it to calculate Yfor anygiven value of G,

without having to run through the entire tedious business of solving the equation. We have saved

ourselves the labour time of solving the equation each and every time the level of government

spending changes.

In the next unit we can see that still more has been achieved: we have established a relationbetween government spending and national income. We shall shortly start using this relation to

determine the effect of government policy on employment.

REFERENCES

Keynes, John Maynard The General Theory of Employment, Interest and MoneyPub:McMillan

This is the work in which Keynes systematically expounded his 'new' doctrine and where you will

find his theories of consumption, liquidity preference and investment behaviour

Hicks, J. R. Mr Keynes and the Classics: a Suggested Interpretation. Econometrica 5:147-159.

1937

This article was the foundation of what has come to be known as the 'neo-classical synthesis'

which has become the standard mathematical interpretation of Keynes's theory.

02-symbols, formulas and rules

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