Earlier in this chapter, we described how autonomous changes in spending—such as a fall in investment spending when a housing bubble bursts—lead to a multi-stage process through the actions of the multiplier that magnifies the effect of these changes on real GDP. In this section, we will examine this multi-stage process more closely. We’ll see that the multiple rounds of changes in real GDP are accomplished through changes in the amount of output produced by firms—changes that they make in response to changes in their inventories. We’ll come to understand why inventories play a central role in macroeconomic models of the economy in the short run as well as why economists pay particular attention to the behaviour of firms’ inventories when trying to understand the likely future state of the economy.

Before we begin, let’s quickly recap the assumptions underlying the multiplier process.

In all subsequent chapters, we will drop the assumption that the aggregate price level is fixed. Appendix 13A addresses how taxes affect the multiplier process. We’ll explain how the interest rate is determined in Chapter 15 and bring foreign trade back into the picture in Chapter 19.

In an economy with no government and no foreign trade, there are only two sources of aggregate expenditure: consumer spending, C, and investment spending, I. And since we assume that there are no taxes or transfers, aggregate disposable income, YD, is equal to nominal GDP (which, since the aggregate price level is fixed, is proportional to real GDP): the total value of final sales of goods and services ultimately accrues to households as income. So in this highly simplified economy, there are two basic equations of national income accounting:

As we learned earlier in this chapter, the aggregate consumption function shows the relationship between disposable income and consumer spending. Let’s continue to assume that the aggregate consumption function is of the same form as in Equation 11-9:

In our simplified model, we will also assume planned investment spending, I_Planned, is autonomous or fixed.

We need one more concept before putting the model together: planned aggregate expenditure, the total amount of planned spending in the economy. Unlike firms, households don’t take unintended actions like unplanned inventory investment. So planned aggregate expenditure is equal to the sum of consumer spending and planned investment spending. We denote planned aggregate expenditure by AE_Planned, so:

Substituting in the aggregate consumption, this becomes:

This can be rewritten as:

where AE₀ represents autonomous planned aggregate expenditure, which collects all of the components of aggregate expenditure that are assumed to be autonomous or fixed. This is the intercept in the planned expenditure line—changes in autonomous expenditure will shift the planned expenditure line in a parallel manner. The marginal propensity to consume is the slope of the planned expenditure line—changes in the MPC will make the planned expenditure line either steeper or less steep.

The level of planned aggregate expenditure in a given year depends on the level of real GDP in that year. To see why, let’s look at a specific example, shown in Table 11-3. We assume that the aggregate consumption function is:

Real GDP, YD, C, I_Planned, and AE_Planned are all measured in billions of dollars, and we assume that the level of planned investment, I_Planned, is fixed at $500 billion per year. The first column shows possible levels of real GDP. The second column shows disposable income, YD, which in our simplified model is equal to real GDP. The third column shows consumer spending, C, equal to $300 billion plus 0.6 times disposable income, YD. The fourth column shows planned investment spending, I_Planned, which we have assumed is $500 billion regardless of the level of real GDP. Finally, the last column shows planned aggregate expenditure, AE_Planned, the sum of aggregate consumer spending, C, and planned investment spending, I_Planned. (To economize on notation, we’ll assume that it is understood from now on that all the variables in Table 11-3 are measured in billions of dollars per year.) As you can see, a higher level of real GDP leads to a higher level of disposable income: every 500 increase in real GDP raises YD by 500, which in turn raises C by 500 × 0.6 = 300 and AE_Planned by 300.

Figure 11-8 illustrates the information in Table 11-3 graphically. Real GDP is measured on the horizontal axis. CF is the aggregate consumption function; it shows how consumer spending depends on real GDP. AE_Planned, the planned aggregate expenditure line, corresponds to the aggregate consumption function shifted up by 500 (the amount of I_Planned). It shows how planned aggregate expenditure depends on real GDP. Both lines have a slope of 0.6, equal to MPC, the marginal propensity to consume.

But this isn’t the end of the story. Table 11-3 reveals that real GDP equals planned aggregate expenditure, AE_Planned, only when the level of real GDP is at 2000. Real GDP does not equal AE_Planned at any other level. Is that possible? Didn’t we learn in Chapter 7, with the circular-flow diagram, that total spending on final goods and services in the economy is equal to the total value of output of final goods and services? The answer is that for brief periods of time, planned aggregate expenditure can differ from real GDP because of the role of unplanned aggregate expenditure—I_Unplanned, unplanned inventory investment. But as we’ll see in the next section, the economy moves over time to a situation in which there is no unplanned inventory investment, a situation called income–expenditure equilibrium. And when the economy is in income–expenditure equilibrium, planned aggregate expenditure on final goods and services equals aggregate output.

For all but one value of real GDP shown in Table 11-3, real GDP is either more or less than AE_Planned, the sum of consumer spending and planned investment spending. For example, when real GDP is 1000, consumer spending, C, is 900 and planned investment spending is 500, making planned aggregate expenditure 1400. This is 400 more than the corresponding level of real GDP. Now consider what happens when real GDP is 2500; consumer spending, C, is 1800 and planned investment spending is 500, making planned aggregate expenditure only 2300, which is 200 less than real GDP.

As we’ve just explained, planned aggregate expenditure can be different from real GDP only if there is unplanned inventory investment, I_Unplanned, in the economy. Let’s examine Table 11-4, which includes the numbers for real GDP and for planned aggregate expenditure from Table 11-3. It also includes the levels of unplanned inventory investment, I_Unplanned, that each combination of real GDP and planned aggregate expenditure implies. For example, if real GDP is 2500, planned aggregate expenditure is only 2300. This 200 excess of real GDP over AE_Planned must consist of positive unplanned inventory investment. This can happen only if firms have overestimated sales and produced too much, leading to unintended additions to inventories. More generally, any level of real GDP in excess of 2000 corresponds to a situation in which firms are producing more than consumers and other firms want to purchase, creating an unintended increase in inventories.

Conversely, a level of real GDP below 2000 implies that planned aggregate expenditure is greater than real GDP. For example, when real GDP is 1000, planned aggregate expenditure is much larger, at 1400. The 400 excess of AE_Planned over real GDP corresponds to negative unplanned inventory investment equal to −400. More generally, any level of real GDP below 2000 implies that firms have underestimated sales, leading to a negative level of unplanned inventory investment in the economy.

By putting together Equations 11-10, 11-11, and 11-14a, we can summarize the general relationships among real GDP, planned aggregate expenditure, and unplanned inventory investment as follows:

So whenever real GDP exceeds AE_Planned, I_Unplanned is positive; whenever real GDP is less than AE_Planned, I_Unplanned is negative.

But firms will act to correct their mistakes. We’ve assumed that they don’t change their prices, but they can adjust their output. Specifically, they will reduce production if they have experienced an unintended rise in inventories or increase production if they have experienced an unintended fall in inventories. And these responses will eventually eliminate the unanticipated changes in inventories and move the economy to a point at which real GDP is equal to planned aggregate expenditure. Staying with our example, if real GDP is 1000, negative unplanned inventory investment will lead firms to increase production (by hiring additional workers if necessary), leading to a rise in real GDP. In fact, this will happen whenever real GDP is less than 2000—that is, whenever real GDP is less than planned aggregate expenditure. Conversely, if real GDP is 2500, positive unplanned inventory investment will lead firms to reduce production (perhaps by laying off some of their workers), leading to a fall in real GDP. This will happen whenever real GDP is greater than planned aggregate expenditure.

The only situation in which firms won’t have an incentive to change output in the next period is when aggregate output, measured by real GDP, is equal to planned aggregate expenditure in the current period, an outcome known as income–expenditure equilibrium. In Table 11-4, income–expenditure equilibrium is achieved when real GDP is 2000, the only level of real GDP at which unplanned inventory investment is zero. From now on, we’ll denote the real GDP level at which income–expenditure equilibrium occurs as Y* and call it the income–expenditure equilibrium (level of) GDP.

Figure 11-9 illustrates the concept of income–expenditure equilibrium graphically. Real GDP is on the horizontal axis and planned aggregate expenditure, AE_Planned, is on the vertical axis. There are two lines in the figure. The solid line is the planned aggregate expenditure line. It shows how AE_Planned, equal to C + I_Planned, depends on real GDP; it has a slope of 0.6, equal to the marginal propensity to consume, MPC, and a vertical intercept equal to A + I_Planned (300 + 500 = 800). The dashed line, which goes through the origin with a slope of 1 (often called a 45-degree line), shows all the possible points at which planned aggregate expenditure is equal to real GDP (i.e., Y = AE_Planned). This line allows us to easily spot the point of income–expenditure equilibrium, which must lie on both the 45-degree line and the planned aggregate expenditure line. So the point of income–expenditure equilibrium is at E, where the two lines cross. And the income–expenditure equilibrium GDP, Y*, is 2000—the same outcome we derived in Table 11-4.

Now consider what happens if the economy isn’t in income–expenditure equilibrium. We can see from Figure 11-9 that whenever real GDP is less than Y*, the planned aggregate expenditure line lies above the 45-degree line and AE_Planned exceeds real GDP. In this situation, I_Unplanned is negative: as shown in the figure, at a real GDP of 1000, I_Unplanned is −400. As a consequence, firms will increase their production and then real GDP will rise. In contrast, whenever real GDP is greater than Y*, the planned aggregate expenditure line lies below the 45-degree line. Here, I_Unplanned is positive: as shown, at a real GDP of 2500, I_Unplanned is 200. The unanticipated accumulation of inventory will lead firms to lay off some of their workers to reduce levels of output, which leads to a fall in real GDP.

The type of diagram shown in Figure 11-9, which identifies income–expenditure equilibrium as the point at which the planned aggregate expenditure line crosses the 45-degree line, has a special place in the history of economic thought. Known as the Keynesian cross, it was developed by Paul Samuelson, one of the greatest economists of the twentieth century (as well as a Nobel Prize winner), to explain the ideas of John Maynard Keynes, the founder of macroeconomics as we know it.

We’ve just learned about a very important feature of the macroeconomy: when planned spending by households and firms does not equal the current aggregate output by firms, this difference shows up as (unplanned) changes in inventories. The response of firms to those inventory changes moves real GDP over time to the point at which real GDP and planned aggregate expenditure are equal. That’s why, as we mentioned earlier, changes in inventories are considered a leading indicator of future economic activity.

Now that we understand how real GDP moves to achieve income–expenditure equilibrium for a given level of planned aggregate expenditure, let’s turn to understanding what happens when there is a shift of the planned aggregate expenditure line. How does the economy move from the initial point of income–expenditure equilibrium to a new point of income–expenditure equilibrium? And what are the possible sources of changes in planned aggregate expenditure?

In our simple model there are only two possible sources of a shift of the planned aggregate expenditure line: a change in planned investment spending, I_Planned, or a shift of the aggregate consumption function, CF. For example, a change in I_Planned can occur because of a change in the interest rate. (Remember, we’re assuming that the interest rate is fixed by factors that are outside the model. But we can still ask what happens when the interest rate changes.) A shift of the aggregate consumption function (that is, a change in its vertical intercept, AC) can occur because of a change in aggregate wealth—say, due to a rise in house prices. When there is a change in autonomous planned aggregate expenditure—when the planned expenditure line shifts (in a parallel manner)—this represents a change in the level of planned aggregate expenditure at any given level of real GDP. Recall from earlier in this chapter that an autonomous change in planned aggregate expenditure is a change in the desired level of spending by firms, households, and government at any given level of real GDP (although we’ve assumed away the government for the time being). How does an autonomous change in planned aggregate expenditure affect real GDP in income–expenditure equilibrium?

Table 11-5 and Figure 11-10 start from the same numerical example we used in Table 11-4 and Figure 11-9. They also show the effect of an autonomous increase in planned aggregate expenditure of 400, which is what happens when planned aggregate expenditure is 400 higher at each level of real GDP. Look first at Table 11-5. Before the autonomous increase in planned aggregate expenditure, the level of real GDP at which planned aggregate expenditure is equal to real GDP, Y*, is 2000. After the autonomous change, Y* has risen to 3000. The same result is visible in Figure 11-10. The initial income–expenditure equilibrium is at E₁, where is 2000. The autonomous rise in planned aggregate expenditure shifts the planned aggregate expenditure line up, leading to a new income–expenditure equilibrium at E₂, where is 3000.

The fact that the rise in income–expenditure equilibrium GDP, from 2000 to 3000, is much larger than the autonomous increase in aggregate expenditure, which is only 400, has a familiar explanation: the multiplier process. In the specific example we have just described, an autonomous increase in planned aggregate expenditure of 400 leads to an increase in Y* from 2000 to 3000, a rise of 1000. So the multiplier in this example is .

We can examine in detail what underlies the multi-stage multiplier process by looking more closely at Figure 11-10. First, starting from E₁, the autonomous increase in planned aggregate expenditure leads to a gap between planned aggregate expenditure and real GDP. This is represented by the vertical distance between X, at 2400, and E₁, at 2000. This gap illustrates an unplanned fall in inventory investment: I_Unplanned = −400. Firms respond by increasing production, leading to a rise in real GDP from . The rise in real GDP translates into an increase in disposable income, YD. That’s the first stage in the chain reaction. But it doesn’t stop there—the increase in YD leads to a rise in consumer spending, C, which sets off a second-round rise in real GDP. This in turn leads to a further rise in disposable income and consumer spending, and so on. And we could play this process in reverse: an autonomous fall in aggregate expenditure will lead to a chain reaction of reductions in production, causing a fall in real GDP and consumer spending.

We can summarize these results in an equation, where ΔAE₀ represents a change in the autonomous portion of planned expenditure, AE_Planned, and , the subsequent change in income–expenditure equilibrium GDP:

Recalling that the multiplier, , is greater than 1, Equation 11-17 tells us that the change in income–expenditure equilibrium GDP, ΔY*, is several times as large as the autonomous change in planned aggregate expenditure, ΔAE₀. It also helps us recall an important point: because the marginal propensity to consume is less than 1, each increase in disposable income and each corresponding increase in consumer spending is smaller than in the previous round. That’s because at each round some of the increase in disposable income leaks out into savings. As a result, although real GDP grows at each round, the increase in real GDP diminishes from each round to the next. At some point the increase in real GDP is negligible, and the economy converges to a new income–expenditure equilibrium GDP at .

The Paradox of Thrift You may recall that in Chapter 6 we mentioned the paradox of thrift to illustrate the fact that in macroeconomics the outcome of many individual actions can generate a result that is different from and worse than the simple sum of those individual actions. In the paradox of thrift, when households and firms anticipate tough(er) economic times, they attempt to save more. These actions depress the economy, leaving households and firms worse off than if they hadn’t acted virtuously to prepare for tough times. It is called a paradox because what’s usually “good” (saving to provide for your family in hard times) is “bad” (because it can make everyone worse off).

Using the multiplier, we can now see exactly how this scenario unfolds. Suppose that there is a slump in consumer spending or investment spending, or both, just like the slump in residential construction investment spending leading up to the 2008–2009 recession. This causes a fall in income–expenditure equilibrium GDP that is several times larger than the original fall in spending. The fall in real GDP leaves consumers and producers worse off than they would have been if they hadn’t cut their spending. Conversely, prodigal behaviour is rewarded: if consumers or producers increase their spending, the resulting multiplier process makes the increase in income–expenditure equilibrium GDP several times larger than the original increase in spending. So prodigal spending makes consumers and producers better off than if they had been cautious spenders.

It’s important to realize that our result that the multiplier is equal to 1/(1 − MPC) depends on the simplifying assumptions that there are no taxes or transfers and that planned investment does not depend directly on GDP, so that disposable income is equal to real GDP. In Appendix 13A, we’ll bring taxes into the picture, which makes the expression for the multiplier more complicated and the multiplier itself smaller. But the general principle we have just learned—an autonomous change in planned aggregate expenditure leads to a change in income–expenditure equilibrium GDP, both directly and through an induced change in consumer spending—remains valid.

As we noted earlier in this chapter, declines in planned investment spending are usually the major factor causing recessions, because historically they have been the most common source of autonomous reductions in aggregate expenditure. The tendency of the consumption function to shift upward over time, which we pointed out earlier in Economics in Action, “Famous First Forecasting Failures,” means that autonomous changes in both planned investment spending and consumer spending play important roles in expansions. But regardless of the source, there are multiplier effects in the economy that magnify the size of the initial change in aggregate expenditure.