Unit 3 Aggregate demand and the multiplier model
3.7 The multiplier model: Aggregate demand shocks cause business cycle fluctuations
- demand shock
- An unexpected or exogenous change in demand. In macroconomics a demand shock means a change in aggregate demand, such as a rise or fall in autonomous consumption, investment, or exports. In microeconomics it refers to an exogenous shift in the demand curve for a particular good. See also: supply shock, exogenous shock.
We can use the same figure to demonstrate the effect of an aggregate demand shock, such as a change in autonomous consumption (\(c_0\)) or investment (\(I\)). First, we will show how the change makes the old outcome no longer an equilibrium, and we then locate the new equilibrium.
Changes in autonomous consumption or investment displace the old equilibrium because they change aggregate demand, which in turn alters the level of output and employment.
An aggregate demand shock: A fall in investment
In Figure 3.15, we take the multiplier diagram and reduce investment. We choose a reduction in investment of €15 billion. Follow the steps in Figure 3.15 to understand what happens. Remember that whenever income falls, aggregate demand falls by 0.6 times the income change.
Figure 3.15 The multiplier in action: an investment-led recession.
Goods market equilibrium
A fall in investment
Firms cut back
A fall in consumption
Firms cut back again
… and so on
The new aggregate demand line
The fall in output as a result of the shock
The multiplier is equal to 2.5
Figure 3.15 traces the successive effects of the fall in investment through the economy. The first-round effect is that the fall in investment cuts aggregate demand by €15 billion. But lower spending also means lower production and lower incomes (firms will lay off workers), leading to a further decline in spending. Think of households where some members lose their job: they would like to keep consumption stable, but when their income falls, they are unable to borrow enough to sustain that level of consumption, so they reduce their spending, which leads to further cuts in production and incomes. The consumption equation tells us that this kind of behaviour leads to a fall in aggregate consumption of 0.6 times the fall in income. This multiplier process will go on until the economy reaches point Z, a recession.
Following the investment shock, the intercept of the line has moved down by €15 billion, causing a parallel shift in the aggregate demand line. Output has fallen by €37.5 billion, much more than the initial fall in investment of €15 billion: this is the multiplier effect.
In this case, the multiplier is equal to 2.5, because the total change in output is 2.5 times larger than the initial change in investment. But a multiplier of 2.5 is unrealistically large. The next section shows that once taxes and imports are introduced in the model, the multiplier shrinks.
- multiplier model
- A model of aggregate demand that includes the multiplier process. See also: fiscal multiplier, multiplier process.
It is this multiplier process that gives the multiplier model of aggregate demand its name. This is a summary:
- A fall in demand leads to a fall in production and an equivalent fall in income: This leads to a further (smaller) fall in demand, which leads to a further fall in production, and so on.
- The multiplier is the sum of all these successive decreases in production: Eventually, output has fallen by a larger amount than the initial change in demand. Output is a multiple of the initial change.
- Production adjusts to demand: Firms supply the amount of goods demanded at the prevailing price. When demand falls, firms adjust production down. The model assumes that they do not adjust their prices.
- The multiplier in this model is always greater than one: Output changes one for one with the initial change in demand; the successive changes add to the initial change.
For the multiplier process to work in the same way in response to a rise in investment, our assumption of spare capacity and fixed wages is important. It means that costs will not rise when output goes up, so firms will be happy to supply the extra output demanded without adjusting their prices. If there was no spare capacity, some of the increased spending would translate into higher prices or wages rather than higher real output—as we discuss in the next unit.
So, if the economy is not characterized by spare capacity and constant wages, the multiplier will be smaller than what we find here.
Question 3.8 Choose the correct answer(s)
Figure 3.15 depicts a consumption function of an economy, where C is the aggregate consumption spending and Y is the current income of the economy.
Based on this information, which of the following statements is correct?
- The MPC is the proportion of the additional income received that is spent on consumption. Here it is given by \(c_1\). \(C/Y\) is the average propensity to consume.
- The slope of the line gives the MPC.
- Households that smooth consumption will choose to increase spending by less than the amount their income increases.
- C is \(c_0 + c_1Y\), that is, $60 trillion plus the fixed (or autonomous) consumption \(c_0\).
Calculating the multiplier, k, and the change in total output
We can show the effect on output and calculate the size of the multiplier by using the equations for the two lines that determine the equilibrium output in the multiplier diagram. The 45-degree line is simply the equation, \(Y\) = AD. Combining this with the equation for AD gives us:
\[Y = \underbrace{c_0 + c_1Y + I}_{\text{aggregate demand}}\]Collecting terms on the left-hand side,
\[\begin{align} Y(1-c_1) = c_0 + I \end{align}\]We then divide through by \((1 − c_1)\):
\[Y = \frac{1}{1-c_1} \times (c_0 + I)\]This equation tells us the equilibrium value of \(Y\) in terms of the MPC and autonomous consumption and investment. For the example drawn in Figure 3.15, where \(c_1=0.6\), \(c_0=10\), and \(I=22\), the equilibrium level of output, \(Y\), is €80 billion.
We will write this equation as:
\[Y=k(c_0+I) \text{ where } k=\frac{1}{1-c_1}\]This equation shows more clearly that equilibrium output, Y, is equal to autonomous demand \((c_0+I)\) multiplied by a factor, k. So if autonomous demand changes, Y will change by k times as much.
This explains why the name given to k is ‘the multiplier’: \(k=\frac{1}{1−c_1}\). You can check that when \(c_1=0.6\), \(k=2.5\).
The Greek letter, \(\Delta\), is often used to denote changes. We read \(\Delta I\) as ‘delta \(I\)’ or ‘the change in investment’.
For example, suppose investment changes by an amount, \(\Delta I\).
The change in output is:
\[\begin{align} \Delta Y &= k(c_0+I+\Delta I)-k(c_0+I) \\ &=k\Delta I \end{align}\]For now, our model only includes two components of aggregate demand (consumption and investment) and the multiplier is equal to \(\frac{1}{1−c_1}\). Once we expand the model to include the government and net exports in Section 3.8, the formula for the multiplier will be a little different because it will reflect the fact that taxes and imports, like consumption, depend on the level of income (output). But in all cases, we denote the multiplier by \(k\), and it tells us the effect on output of an exogenous demand shock.
Try this Excel multiplier simulation to investigate how changing the marginal propensity to consume affects the size and duration of an economic shock.
Calculating the multiplier
We consider the effect of an increase in investment of €15 billion. We can summarize our findings from the multiplier diagram by doing some algebra. To get the multiplier, we can calculate the total increase in production after \(n + 1\) rounds of the process. Each round of the process matches the circular flow diagram. The first-round increase in demand and production is €15 billion. The second-round increase in demand and production is \((c_1 × \text{€15 billion})\), the third-round increase in demand and production is \(c_1 × (c_1 × \text{ €15 billion}) = (c_1^2 × \text{ €15 billion})\), and so on.
Following this logic, the total increase in demand and production after \(n + 1\) rounds is the total sum of these changes:
\[15 + c_1(15) + c_1^2(15) + … + c_1^n(15) = 15(1+ c_1 + c_1^2 + … + c_1^n)\]Because the marginal propensity to consume is lower than one, we can show that the total sum in the brackets reaches a limit of \(1/(1 − c_1)\) as \(n\) gets large. This is because the term in the brackets is, mathematically, a geometric series. (A geometric series is the sum of an infinite number of terms that have a common ratio between each term, in this case, the marginal propensity to consume, \(c_1\).) We show this result as follows.
If \(k\) is the multiplier, we have:
\[\begin{align} k = (1+c_1 + c_1^2 + … + c_1^n) \end{align}\]Now multiply both sides by \((1 − c_1)\) to get:
\[\begin{align} k(1−c_1) &= (1 + c_1 + c_1^2 + … + c_1^n)(1−c_1) \\ &= (1 + c_1 + c_1^2 + … + c_1^n) - (c_1 + c_1^2 + c_1^3 + … + c_1^{n+1}) \\ &= 1 − c_1^{n+1} \end{align}\]Now divide again by \((1 − c_1)\):
\[\begin{align} k = \frac{(1 − c_1^{n+1})}{(1 − c_1)} \end{align}\]As \(n\) gets large, assuming \(c_1 < 1\), the numerator tends to 1. So, in the limit:
\[\begin{align*} k = \frac{1}{1-c_1} \end{align*}\]In the example, the marginal propensity to consume is, on average, 0.6. This implies that the multiplier is equal to:
\[\begin{align} \frac{1}{1−c_1} = \frac{1}{1-0.6} = 2.5 \end{align}\]We can then apply the multiplier to the initial change in investment of €15 billion to find the sum of all the successive increases in production triggered by the initial hike in investment and aggregate demand: \(2.5 × \text{ €15 billion} = \text{ €37.5 billion}\).
Question 3.9 Choose the correct answer(s)
The following diagram depicts the change in the aggregate goods market equilibrium when there is a €2 billion increase in investment.
The economy’s marginal propensity to consume is 0.5. Based on this information, which of the following statements is correct?
- The equilibrium is where the aggregate consumption line intersects with the 45-degree line. Therefore the new equilibrium is Z.
- The diagram shows that the €2 billion increase in investment results in a €4 billion increase in aggregate demand.
- The multiplier is equal to \(1/(1 − 0.5) = 2\).
- The distance between A and B is the initial increase in investment of €2 billion. At C, the output Y has increased by €2 billion compared with B, which results in an increase in the aggregate demand of ΔY (change in Y) times MPC, that is, \(\text{€2 billion} × 0.5 = \text{€1 billion}\).
