THINKING LIKE AN ECONOMIST
optimism in the area causes long-term investment as firms have confidence in an expanding market. This encourages house building and other forms of investment in infrastructure and services. And so the region thrives.
Meanwhile the declining region suffers from deprivation as unemployment rises. This encourages people to move away and businesses to close. There is a further decline in jobs and further migration from the region.
Cumulative causation does not just occur at a macro level. If a company is successful, it is likely to find raising extra finance easier; it may be able to use its power more effectively to out-compete rivals. Giant companies, such as Microsoft, can gain all sorts of economies of scale, including network economies (see Case Study 6.4 in MyEconLab), all of which help the process of building their power base. Success breeds success.
1. How might cumulative causation work at the level of an individual firm that is losing market share?
2. Are there any market forces that work against cumulative causation? For instance, how might markets help to arrest the decline of a depressed region of the economy and slow down the expansion of a booming region
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Once an economy starts to expand, growth is likely to gather pace. Once it starts slowing down, this can gather pace too and end up in a recession. There are many other examples in economics of things getting ‘onto a roll’. A ris- ing stock market is likely to breed confidence in investors and encourage them to buy. This ‘destabilising speculation’
(see pages 73–4) will then lead to further rises in share prices. A fall in stock market prices can lead to panic sell- ing of shares. The booming stock market of the late 1990s and the falls in the early 2000s are good examples of this.
This phenomenon of things building on themselves is known as ‘cumulative causation’ and occurs throughout market economies. It is a threshold conceptbecause it helps us to understand the built-in instability in many parts of the economy and in many economic situations.
Central to explaining cumulative causation is people’s psychology. Good news creates confidence and this optim- ism causes people to behave in ways that build on the good news. Bad news creates pessimism and this leads to people behaving cautiously, which tends to reinforce the bad news.
Take two regions of an economy: an expanding region and a declining region. The expansion of the first region encourages workers to move there in search of jobs. The
the W function is given by the marginal propensity to withdraw (ΔW/ΔY). The less steep the line (and hence the lower the mpw), the bigger will be the rise in national income: the bigger will be the multiplier.
Try this simple test of the above argument. Draw a series of W lines of different slopes, all crossing the J line at the same point. Now draw a second J line above the first.
Mark the original equilibrium and all the new ones corresponding to each of the W lines. It should be quite obvious that the flatter the W line is, the more Y will have increased.
The point here is that the less is withdrawn each time extra income is generated, the more will be recirculated and hence the bigger will be the rise in national income.
The size of the multiplier thus varies inversely with the size of the mpw. The bigger the mpw, the smaller the multiplier;
the smaller the mpw, the bigger the multiplier. In fact, the multiplier formulasimply gives the multiplier as the inverse of the mpw:
k =1/mpw
or alternatively, since mpw + mpcd =1 and thus mpw = 1 −mpcd,
k=1/(1 −mpcd)
Thus if the mpwwere 1/4(and hence the mpcdwere 3/4), the multiplier would be 4. So if Jincreased by £10 billion, Y would increase by £40 billion.
But why is the multiplier given by the formula 1/mpw?
This can be illustrated by referring to Figure 17.8. The mpw is the slope of the W line. In the diagram, this is given by the amount (b −c)/(c −a). The multiplier is defined as
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A shift in withdrawals
A multiplied rise in income can also be caused by a fall in withdrawals. This is illustrated in Figure 17.9.
The withdrawals function shifts from W1 to W2. This means that, at the old equilibrium of Ye1, injections now exceed withdrawals by an amount a −b. This will cause national income to rise until a new equilibrium is reached ΔY/ΔJ. In the diagram, this is the amount (c −a)/(b −c). But this is merely the inverse of the mpw. Thus the multiplier equals 1/mpw.1
Figure 17.8 The multiplier: a shift in injections
1In some elementary textbooks, the formula for the multiplier is given as 1/mps. The reason for this is that it is assumed (for simplicity) that there is only one withdrawal, namely saving, and only one injection, namely investment. As soon as this assumption is dropped, 1/mpsbecomes the wrong formula.
*LOOKING AT THE MATHS
The multiplier can be expressed as the first derivative of national income with respect to injections.
k=
Since in equilibrium J =W, it is also the first derivative of income with respect to withdrawals. Thus
k=
The marginal propensity to withdraw (i.e. the slope of the withdrawals curve) is found by differentiating the withdrawals function:
mpw= =
Thus k=
The algebra of the multiplier is explored in MyEconLab in Maths Case 17.1, which does not use calculus, and Maths Case 17.2, which does.
1 mpw
1 k dW dY dY dW dY
dJ
(Injections) multiplier formula The formula for the multiplier: k=1/mpwor 1/(1 −mpcd).
Definition
Figure 17.9 The multiplier: a shift in withdrawals
at Ye2where J=W2. Thus a downward shift of the with- drawals function of a − b(ΔW) causes a rise in national income of c −a(ΔY). The multiplier in this case is given by ΔY/ΔW: in other words, (c −a)/(a −b). Note that the multi- plier is based on the initialfall in withdrawals. Once the multiplier effect has worked through, withdrawals will have risen back to equal injections at point c.
Why is the ‘withdrawals multiplier’ strictly speaking a negative figure?
The multiplier: the income and expenditure approach
The multiplier can also be demonstrated using the income/expenditure approach. Assume in Figure 17.10 that the aggregate expenditure function shifts to E2. This could be due either to a rise in one or more of the three injec- tions, or to a rise in the consumption of domestically pro- duced goods (and hence a fall in withdrawals). Equilibrium national income will rise from Ye1to Ye2.
What is the size of the multiplier? The initial rise in expenditure was b −a. The resulting rise in income is c −a.
The multiplier is thus (c −a)/(b −a).
The effect is illustrated in Table 17.2. Consumption of domestic product (Cd) is shown in column 2 for various
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levels of national income (Y). For every £100 billion rise in Y, Cdrises by £80 billion. Thus the mpcd=0.8. Assume initially that injections equal £100 billion at all levels of national income. Aggregate expenditure (column 4) equals Cd+J. Equilibrium national income is £700 billion. This is where Y=E.
Now assume that injections rise by £20 billion to
£120 billion. Aggregate expenditure is now shown in the final column and is £20 billion higher than before at each level of national income (Y). At the original equilibrium national income (£700 billion), aggregate expenditure is now £720 billion. This excess of Eover Yof £20 billion will generate extra incomes and continue doing so as long as E remains above Y. Equilibrium is reached at £800 billion, where once more Y =E. The initial rise in aggregate expen- diture of £20 billion (from £700bn to £720bn) has led to an eventual rise in both national income and aggregate expen- diture of £100 billion. The multiplier is thus 5 (i.e. £100bn/
£20bn). But this is equal to 1/(1 −0.8) or 1/(1 −mpcd).
1. What determines the slope of the E function?
2. How does the slope of the E function affect the size of the multiplier? (Try drawing diagrams with E functions of different slopes and see what happens when they shift.)
The multiplier: a numerical illustration
The multiplier effect does not work instantaneously. When there is an increase in injections, whether investment, government expenditure or exports, it takes time before this brings about the full multiplied rise in national income.
Consider the following example. Let us assume for sim- plicity that the mpwis 1/2. This will give an mpcdof 1/2also.
Let us also assume that investment (an injection) rises by
£160 million and stays at the new higher level. Table 17.3 shows what will happen.
As firms purchase more machines and construct more factories, the incomes of those who produce machines and those who work in the construction industry will increase by £160 million. When this extra income is received by households, whether as wages or profits, half will be with- drawn (mpw=1/2) and half will be spent on the goods and services of domestic firms. This increase in consumption
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Figure 17.10 The multiplier: a shift in the expenditure function
Table 17.2 The effect of an increase in aggregate expenditure (£ billions)
Y
Y CCdd JJ ((oolldd)) EE ((oolldd)) JJ ((nneeww)) EE ((nneeww))
500 440 100 540 120 560
600 520 100 620 120 640
700 600 100 770000 120 720
800 680 100 780 120 880000
900 760 100 860 120 880
Table 17.3 The multiplier ‘round’
Round ΔΔJ (£m) ΔΔY (£m) ΔΔCd(£m) ΔΔW (£m)
1 160 160 80 80
2 – 80 40 40
3 – 40 20 20
4 – 20 10 10
5 – 10 5 5
6 – 5 ã ã
. . . .
1 → ∞ 320 160 160
thus generates additional incomes for firms of £80 million over and above the initial £160 million (which is still being generated in each time period). When this additional
£80 million of incomes is received by households (round 2), again half will be withdrawn and half will go on consumption of domestic product. This increases national income by a further £40 million (round 3). And so each time we go around the circular flow of income, national income increases, but by only half as much as the previous time (mpcd=1/2).
If we add up the additional income generated in each round (assuming the process goes on indefinitely), the total will be £320 million: twice the rise in injections. The multi- plier is 2.
The bigger the mpcd(and hence the smaller the mpw), the more will expenditure rise each time national income rises, and hence the bigger will be the multiplier.
*The multiplier: some qualifications
(This section examines the multiplier formula in more detail. You may omit it without affecting the flow of the argument.)
Some possible errors can easily be made in calculating the value of the multiplier. These often arise from a confu- sion over the meaning of terms.
The marginal propensity to consume domestic product
Remember the formula for the multiplier:
k=1/(1 −mpcd)
It is important to realise just what is meant by the mpcd. It is the proportion of a rise in households’ gross (i.e. pre- tax-and-benefit) income that actually accrues to domestic firms. It thus excludes that part of consumption that is spent on imports and that part which is paid to the govern- ment in VAT and other indirect taxes.
Up to now we have also been basing the mpcon gross income. As Case Study 17.1 in MyEconLab shows, however, the mpc is often based on disposable (i.e. post-tax-and-
benefit) income. After all, when consumers decide how much to spend, it is their disposable income rather than their gross income that they will consider. So how do we derive the mpcd (based on gross income) from the mpc based on disposable income (mpc′)? To do this, we must use the following formula:
mpcd=mpc′(1 −tE)(1 −tY) −mpm
where tYis the marginal rate of income tax, and tEis the marginal rate of expenditure tax.
To illustrate this formula consider the following effects of an increase in national income of £100 million. It is assumed that tY=20 per cent, tE=10 per cent and mpc=7/8.
It is also assumed that the mps(from gross income) =1/10 and the mpm(from gross income) =13/100. Table 17.4 sets out the figures.
Gross income rises by £100 million. Of this, £20 million is taken in income tax (tY=20 per cent). This leaves a rise in disposable income of £80 million. Of this, £10 million is saved (mps=1/10) and £70 million is spent. Of this, £7 mil- lion goes in expenditure taxes (tE=10 per cent) and £13 million leaks abroad (mpm=13/100). This leaves £50 mil- lion that goes on the consumption of domestic product (mpcd=50/100 =1/2). Substituting these figures in the above formula gives:
mpcd=mpc(1 −tE)(1 −tY) −mpm
=–78(1 −––101)(1 −––102) −––10013
=(–78×––109 ×––108) −––10013
=100––63−––10013=––10050=–12
Table 17.4 Calculating the mpcd
ΔY – ΔTY = ΔYdis
(£m) 100 20 80
ΔYdis – ΔS = ΔC
(£m) 80 10 70
ΔC – ΔTE – ΔM = ΔCd
(£m) 70 7 13 50
BOX 17.4 DERIVING THE MULTIPLIER FORMULA An algebraic proof
EXPLORING ECONOMICS
But in equilibrium we know that W=J. Hence any change in injections must be matched by a change in withdrawals and vice versa, to ensure that withdrawals and injections remain equal. Thus
ΔW= ΔJ (4)
Substituting equation (4) in equation (3) gives 1/mpw= ΔY/ΔJ (=k)
i.e. the multiplier equals 1/mpw.
The formula for the multiplier can be derived using simple algebra. First of all, remember how we defined the multiplier:
k≡ ΔY/ΔJ (1)
and the marginal propensity to withdraw:
mpw≡ ΔW/ΔY (2)
If we now take the inverse of equation (2), we get
1/mpw≡ ΔY/ΔW (3)
Note that the mpcd, mps, mpmand mptare all based on the rise in gross income, not disposable income. They are 50/100, 10/100, 13/100 and 27/100 respectively.
Maths Case 17.3 in MyEconLab derives the multiplier formula when the propensities to consume, save and import are all based on disposableas opposed to gross income.
Assume that the rate of income tax is 15 per cent, the rate of expenditure tax is 12–12per cent, the mps is––201, the mpm is –18and the mpc (from disposable income) is––1617. What is the mpcd? Construct a table like Table 17.4 assuming again that national income rises by £100 million.
The effects of changes in injections and
withdrawals on other injections and withdrawals
In order to work out the size of a multiplied rise or fall in income, it is necessary to know first the size of the initial totalchange in injections and/or withdrawals. The trouble is that a change in one injection or withdrawal can affect others. For example, a rise in income taxes will reduce not only consumption, but also saving, imports and the rev- enue from indirect taxes. Thus the total rise in withdrawals will be lessthan the rise in income taxes.
Give some other examples of changes in one injection or withdrawal that can affect others.
The relationship between the 45° line diagram and the aggregate demand and supply diagram
We have used two diagrams to show the determination of equilibrium national income: the aggregate demand and supply diagram and the 45° line diagram. The first shows aggregate demand dependent on the price level. The second shows aggregate demand (i.e. aggregate expenditure (E)) dependent on the level of national income. Figure 17.11 shows the multiplier effect simultaneously on the two diagrams. Initially, equilibrium is at Ye1 where aggregate
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demand equals aggregate supply and where the aggregate expenditure line crosses the 45° line.
Now assume that there is an autonomous increase in expenditure. For example, increased business confidence results in increased investment. In Figure 17.11(b), the E line shifts to E2. There is a multiplied rise in income to Ye2. In Figure 17.11(a), the aggregate supply curve is drawn as a horizontal straight line between Ye1and Ye2. This means that an increase in aggregate demand from AD1to AD2will raise income to Ye2with no increase in prices.
But what if the economy is approaching full employ- ment? Surely we cannot expect the multiplier process to work in the same way as when there is plenty of slack in the economy? In this case, the aggregate supply curve will be upward sloping. This means that an increase in aggregate demand will raise prices and not just output. How do we analyse this with the 45° line diagram? We examine this in the next section.
TC 15 p487
Figure 17.11 Showing the multiplier effect on the 45° line and AD/ASdiagrams
Section summary
1. Equilibrium national income can be shown on the 45°
line diagram at the point where W=Jand Y=E.
2. If there is an increase in injections (or a reduction in withdrawals), there will be a multiplied rise in national income. The multiplier is defined as ΔY/ΔJ.
3. The size of the multiplier depends on the marginal propensity to withdraw (mpw). The smaller the mpw, the less will be withdrawn each time incomes are generated round the circular flow, and thus the more will go round again as additionaldemand for domestic product. The multiplier formula is k=1/mpwor 1/(1 −mpcd).
*4. When working out the size of the multiplier, you must be careful to identify clearly the mpcd(which is based on grossincome and only includes expenditure that actually accrues to domestic firms) and not to confuse it with the mpcbased on disposableincome (which includes consumption of imports and the payment of indirect taxes). It is also necessary to identify the fullchanges in injections and withdrawals on which any multiplier effect is based.
5. The multiplier effect can also be illustrated on an aggregate demand and supply diagram.
KI 12 p84
BOX 17.5 THE PARADOX OF THRIFT When prudence is folly
EXPLORING ECONOMICS
about ‘underconsumption’ that had been made back in the sixteenth and seventeenth centuries:
In 1598 Laffemas . . . denounced the objectors to the use of French silks on the grounds that all purchasers of French luxury goods created a livelihood for the poor, whereas the miser caused them to die in distress.
In 1662 Petty justified ‘entertainments, magnificent shews, triumphal arches, etc.’, on the ground that their costs flowed back into the pockets of brewers, bakers, tailors, shoemakers and so forth . . . In 1695 Cary argued that if everybody spent more, all would obtain larger incomes ‘and might then live more plentifully’.1
But despite these early recognitions of the danger of underconsumption, the belief that saving would increase the prosperity of the nation was central to classical economic thought.
Is an increase in saving ever desirable?
1J. M. Keynes, The General Theory of Employment, Interest and Money (Macmillan, 1967), pp. 358–9.
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The classical economists argued that saving was a national virtue. More saving would lead via lower interest rates to more investment and faster growth. Keynes was at pains to show the opposite. Saving, far from being a national virtue, could be a national vice.
Remember the fallacy of composition (see Box 3.7 on page 84). Just because something is good for an individual, it does not follow that it is good for society as a whole. This fallacy applies to saving. If individuals save more, they will increase their consumption possibilities in the future. If society saves more, however, this may reduce its future income and consumption. As people save more, they will spend less. Firms will thus produce less. There will thus be a multiplied fallin income. The phenomenon of higher saving leading to lowernational income is known as ‘the paradox of thrift’.
But this is not all. Far from the extra saving encouraging more investment, the lower consumption will discourage firms from investing. If investment falls, the Jline will shift downwards. There will then be a further multiplied fall in national income. (This response of investment to changes in consumer demand is examined in section 17.4 under the ‘accelerator theory’.)
The paradox of thrift had in fact been recognised before Keynes, and Keynes himself referred to various complaints
‘Full-employment’ national income
The simple Keynesian theory assumes that there is a max- imum level of national output, and hence real income, that can be obtained at any one time. If the equilibrium level of income is at this level, there will be no deficiency of aggregate demand and hence no disequilibrium un- employment. This level of income is referred to as the full- employment level of national income. (In practice, there would still be some unemployment at this level because of equilibrium unemployment – structural, frictional and seasonal.)
Governments of the 1950s, 1960s and early 1970s aimed to achieve this full-employment income (YF), if inflation and the balance of payments permitted. To do this, they attempted to manipulate the level of aggregate demand.
This approach was also adopted by the George W. Bush
administration in the USA in 2001 and again in 2007/8, when attempts were made to stimulate aggregate demand through the use of fiscal and monetary policy in order to stimulate the US economy and prevent it falling into recession. These policies were then used in earnest in late 2008 and in 2009 by many countries around the world to combat a deepening recession. For example, the incoming Obama administration introduced a massive stimulus package of $787 billion (5.7 per cent of US GDP).
The deflationary gap
If the equilibrium level of national income (Ye) is below the full-employment level (YF), there will be excess capacity in the economy and hence demand-deficient unemployment.
There will be what is known as a deflationary orreces- sionary gap. This is illustrated in Figure 17.12.