The derived unique equilibrium income depends on planned spending. An im-portant question to follow from this is, how does equilibrium income change if one of the actors involved revises spending plans?
Assume that the government raises expenditure by ΔG. Then income obvi-ously also rises by ΔG. But this cannot be the end of the story. Households will
Y0 Y0
Y0 Y0
The Keynesian crossis a diagram which plots planned expenditure against income and actual expenditure against income. Equilibrium income obtains where both lines cross.
Newequilibrium
Old equilibrium
Round #1
Round #2
Income
Expenditure
Y0 Y1
AENEW 45°
AEOLD 1
A
A' c
c c
1
1/(1-c)
c2
c2 c2
ΔG = 1
Figure 2.11 If government spending or any other autonomous spending increases by 1, the AE line shifts up by 1. Income rises by due to the multiplier effect. The rea-soning behind this is that the government’s spending increase of 1 not only raises income by 1, leading to point A. It also induces added consumption of c, adding c to income. This raises consumption and income by another , and so on. All these effects add up to the multiplier¢Y = 1>(1 - c). c2
¢Y 7 1
want to spend part of this additional income: this added consumer spending again adds to income, which raises consumer spending even further, and so on. This process is complicated and worth looking at in more detail with the help of Figure 2.11 and Table 2.3.
Let the economy initially be in equilibrium at the point labelled A. Now the government increases expenditure per period by one unit . We will trace the consequences of this spending increase through a series of fictitious rounds, in order to finally arrive at the cumulative total effect. These rounds may have a time dimension – say, months, because consumers respond to in-come increases with a lag of one month. Or we may consider the division of the total effect into a number of rounds as simply a conceptional tool to guide the analysis, and the total effect may actually accrue instantaneously thanks to the foresight of firms.
In round #1, one additional unit of government expenditure shifts the aggre-gate-expenditure line upwards by one unit. This constitutes an increase of
(¢G = 1)
2.2 Income determination: a first look 47
At equilibrium incomeall spending is planned spending (or income equals aggregate expenditure).
aggregate spending at the old income level by one unit, and also raises in-come by one unit to . In round #2, consumers plan to spend the fraction c of their first-round income increase, also raising income by c. By now the total income increase adds up to , and already exceeds the increase in govern-ment spending which triggered this process. The process continues, however. In round #3, consumers add a fraction c of their second-round income increase of c to their old level of spending, meaning that they increase consumption by . Income increases by as well. By analogous reasoning, consump-tion and income increase by in round #4, by in round #5 and so on.
Does this process ever end? Yes and no. With respect to time or rounds, it does go on forever, but the income increases that accrue in each new round will eventually peter out. This is because the marginal propensity to consume c is between zero and one. Raising c to the power of a higher and higher num-ber yields smaller and smaller results. If the second-round effect was , the effect in round #11 is already reduced to . This guarantees that the total effect of a one-unit increase of government spending does not grow infinitely large, despite the fact that an algebraic expression for this com-prises infinitely many terms:
(2.6) But then how large is the total effect on income? One way to figure this out is by noting that in equilibrium
Equilibrium condition Substituting the consumption function (2.5) for C gives
Subtracting cY from both sides and dividing by yields an expression for equilibrium incomeas a function of all autonomous expenditure:
Equilibrium income (2.7)
income Y = aggregate expenditure AE 1 + c + c2 + c3 + c4 + Á + cq
Table 2.3 Income effects of an incease of government expenditure. The table traces the income gains generated by a one-unit increase in government spending. In round #1, income only rises by 1 as output follows demand. In round #2, individuals want to consume a fraction c out of their income increase experienced in round #1. This raises output and in-come by c as well, bringing the total effect to (fifth column). This process continues.
The initial demand increase multiplies into more and more additions to demand in subse-quent rounds. Note, however, that these additions become smaller and smaller.
Round ¢G ¢C
(summed over all rounds)
¢Y
Maths note. We have hit upon a rule that has many uses in economics: for any parameter the series
converges to
as the exponent grows to infinity. Or, more precisely,
. As and , the effects listed in equation (2.6) and in Table 2.2 constitute such a series.
c1 =c
Income only changes if either G, I or NX changes:
(2.8) In our present example we have . Substituting this into equation (2.8) and dividing by gives us
Multiplier (2.9) An increase of government expenditure (or investment, or net exports) by 1 multiplies into an income increase by . This is why the expression given on the right-hand side is called the multiplier. Its numerical value de-pends on the value of c. For the multiplier is 5. For the multi-plier is 10. The verbal explanation is that additional government spending initially makes firms produce just that much more. To be able to do that, they need to employ more labour and pay for it and so income rises. Part of this income returns to the firms as consumption demand. So more labour must be hired and income rises further. When this process comes to a halt, income has risen by the multiplier times the initial increase in government spending.
Additional insight into the economics behind the multiplier is obtained by giving it a circular-flow-of-income interpretation. We recall that some initial, exogenous increase in demand, such as a rise in government spending, not only generates an increase in output and income to match. Rather, this generated crease in income gives rise to even higher demand and, hence, even higher in-come, and thus larger demand still, and so on. In terms of the circular flow of income, the multiplier exists only to the extent that at least a fraction of income eventually returns to firms in the form of demand for domestically produced goods and services. The larger this fraction is, the larger is the multiplier. How does this correspond to our multiplier formula ?
Note that by making use of the identity , the multiplier formula rewrites . The number 1 in the numerator is the presumed increase in gov-ernment spending – an injection of €1 into the income circle. The denomina-tor, s, is the fraction of the injected demand that leaks out of the income circle.
So if the entire injection leaks out of the income circle, that is if , the multiplier is down to 1. Hence, the rise in government spending does not trig-ger any additional spending at all. (In this case the aggregate-expenditure line in Figure 2.10 would be horizontal.) At the other extreme, if none of the added government spending leaks from the income circle, because s drops to zero, the multiplier approaches infinity. (Now the AE line would have a slope of 1.) This leads to the general rule that the multiplier becomes smaller when the leakages out of the income circle generated by any income gain become larger.
This rule will prove useful when we look at the multiplier in the context of a more refined model.
What has fascinated previous generations of economists (and politicians) about the multiplier is that by raising spending, governments can induce in-come increases far greater than the initial spending increase. Before we get too excited about this result, however, let me caution that many of the refinements to be discussed in the remaining sections of this chapter and in subsequent chapters will gradually erode the quantitative importance of the multiplier.
s = 1
Rule.The multiplier is small when a large part of any increase in income leaks out of the circular flow.
The multipliermeasures the income change resulting from a one-unit increase in autonomous expenditure.
2.3 Income determination: a second look 49
Disposable incomeis that part of income left to households after the payment of taxes.
The marginal income tax ratesays by how much taxes rise if income rises by one unit. The average income tax rategives the share of taxes on income on average, that is T/Y. In equation (2.11) the marginal and the average income tax rate are the same.
1
A
c
1
c (1-t)-m 1
Expenditure 1 exceeds output
Output, income
Expenditure
Y0
Equilibrium income Y0
Actual
expenditure First AE line Flatter AE line
Output exceeds expenditure
Figure 2.12 If taxes and imports rise with income, the leaks out of the circular flow increase as income rises. Hence, aggregate expenditure does not increase with income as fast as it did in Figure 2.10. This is repre-sented by the flatter AE line, which now has slope . A flatter AE line signals a smaller multiplier effect. (Note that the flatter line is drawn for higher autonomous spending. Otherwise both lines would intersect on the vertical axis, not at A.)
c(1 - t) - m
2.3 Income determination: a second look
Now that we understand the basic concepts of equilibrium income and the multiplier we move on to a more realistic scenario. This perspective includes more plausible behavioural and institutional features, and then introduces expectations. We begin by deriving a refined but traditional version of the multiplier.
The previous analysis has ignored taxes T. When individuals pay taxes, this reduces gross income Y to disposable income . Individuals are free only to decide how to split disposable income between consumption and savings.
Hence
Consumption function (2.10) In the simplest case, taxes can be thought of as being proportional to income, Tax equation (2.11) where t is the marginal and average income tax rate.
For a first look at imports IM, which we shall refine later, let these also be proportional to income.
Import function (2.12) where m denotes the marginal propensity to import. We continue to consider the remaining components of demand, I, G and EX, exogenous.
This refined model of aggregate demand yields a much flatter AE line than the previous model, as Figure 2.12 illustrates. An equation for the AE line is obtained by substituting equations (2.10), (2.11) and (2.12) into (2.1). After collecting terms this gives
Aggregate expenditure (2.13) The slope of the flatter AE line is due to two effects. First, income-dependent taxes reduce disposable income to ( )Y even before individuals decide whether to save or consume. Consumption out of each unit
1 - t c(1 - t) - m
AE = [c(1 - t) - m]Y + I + G + EX IM = mY
T = tY C = c(Y - T)
Y - T
of income then is only . Second, m out of each unit of income leaks abroad in payment for imports, so only out of each additional unit of income is used to buy domestically produced goods.
An algebraic expression for equilibrium income in the refined model is ob-tained by requiring aggregate spending (as shown in equation (2.13)) to equal income. After solving for Y this yields
Equilibrium income The new multiplier is obtained by taking first differences, letting
and dividing by G:
Multiplier (2.14) Substituting some plausible numbers into the multiplier equations (2.9) and (2.14) illustrates that the achieved refinement has important quantitative con-sequences. With a marginal propensity to consume of , the simple mul-tiplier given in equation (2.9) stands at a value of 10. Now substitute the same value for c into the refined multiplier given in equation (2.14), assume a mod-est tax rate of , let the marginal propensity to import be about one-fifth , and you will arrive at a multiplier of 2.
Why has the multiplier shrunk? Remember our rule: the multiplier becomes smaller when the leakages out of the income circle that result from a given in-come hike do grow. More precisely and in general terms: the multiplier is the initial increase in demand, the injection, divided by the added leakages. By mak-ing taxes and imports dependent on income we have added two more channels through which the initial increase in income, triggered by the government spending hike, leaks out of the income circle. This becomes visible if we make use of the identity to rewrite the refined multiplier from (2.14) as
Each term in the denominator refers to one particular leakage. Starting from the right, when government spending rises by€1, raising output and income by €1 as well (the number in the numerator), a fraction m leaks abroad, a fraction t goes to the government, and a fraction is being saved, which is that part of income left to the disposal of households, , times the savings rate, s.
Equation (2.14) also shows that fiscal policy is not restricted to the manip-ulation of government spending. A second instrument of fiscal policy is the tax rate, which it can change independently of government spending, if so desired.
If the government decreases the marginal rate of income taxation, disposable income increases, the multiplier increases and the AE line becomes steeper (see Figure 2.14). Equilibrium income rises from to .
We conclude this chapter by discussing consumption and investment de-mand from a more modern intertemporal perspective. The payoff will be a first encounter with expectations, which play a prominent role in modern economic analysis, and a first insight into when it is safe to work with the multiplier model, and when not to.
Y1 Y0 1 - t s(1 - t)
¢Y
¢G =
1
s(1 - t) + t + m c = 1 - s (m = 0.22)
t = 0.2
c = 0.9
¢Y
¢G =
1
1 - c(1 - t) + m 0
¢I = ¢EX =
Y = 1
1 - c(1 - t) + m(G + I + EX)
c(1 - t) - m c(1 - t)
Empirical note. Empirical estimates of multipliers for industrial countries range between values of 1.5 and 3.
2.3 Income determination: a second look 51