1.3 Module 26: The Income–Expenditure Model

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Using the Income–Expenditure Model

Earlier in this section, we described how autonomous changes in spending—such as a fall in investment spending when a housing bubble bursts—lead to a multistage process through the actions of the multiplier that magnifies the effect of these changes on real GDP. In this module, we will examine this multistage 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 behavior 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.

1. Changes in overall spending lead to changes in aggregate output. We assume that producers are willing to supply additional output at a fixed price level. As a result, changes in spending translate into changes in output rather than moves of the overall price level up or down. A fixed aggregate price level also implies that there is no difference between nominal GDP and real GDP. So we can use the two terms interchangeably in this module.
2. The interest rate is fixed. As we’ll see, the model we examine here can still be used to study the effects of a change in the interest rate.
3. Taxes, government transfers, and government purchases are all zero.
4. Exports and imports are both zero.

Planned Aggregate Spending and Real GDP

In an economy with no government and no foreign trade, there are only two sources of aggregate spending: consumer spending, C, and investment spending, I. And since we assume that there are no taxes or transfers, aggregate disposable income is equal to GDP (which, since the aggregate price level is fixed, is the same as 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 in the previous module, 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 25-5:

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

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

The level of planned aggregate spending 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 26-1. 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 spending, 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 26-1 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 26-1 illustrates the information in Table 26-1 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 spending line, corresponds to the aggregate consumption function shifted up by 500 (the amount of I_Planned). It shows how planned aggregate spending depends on real GDP. Both lines have a slope of 0.6, equal to MPC, the marginal propensity to consume.

The lower line, CF, is the aggregate consumption function constructed from the data in Table 26-1. The upper line, AE_Planned, is the planned aggregate spending line, also constructed from the data in Table 26-1. It is equivalent to the aggregate consumption function shifted up by $500 billion, the amount of planned investment spending, I_Planned.

But this isn’t the end of the story. Table 26-1 reveals that real GDP equals planned aggregate spending, AE_Planned, only when the level of real GDP is at 2,000. Real GDP does not equal AE_Planned at any other level. Is that possible? Didn’t we learn from 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 spending can differ from real GDP because of the role of unplanned aggregate spending—I_Unplanned, unplanned inventory investment. But as we’ll see, 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 spending on final goods and services equals aggregate output.

Income–Expenditure Equilibrium

For all but one value of real GDP shown in Table 26-1, 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 1,000, consumer spending, C, is 900 and planned investment spending is 500, making planned aggregate spending 1,400. This is 400 more than the corresponding level of real GDP. Now consider what happens when real GDP is 2,500; consumer spending, C, is 1,800 and planned investment spending is 500, making planned aggregate spending only 2,300, 200 less than real GDP.

As we’ve just explained, planned aggregate spending can be different from real GDP only if there is unplanned inventory investment, I_Unplanned, in the economy. Let’s examine Table 26-2, which includes the numbers for real GDP and for planned aggregate spending from Table 26-1. It also includes the levels of unplanned inventory investment, I_Unplanned, that each combination of real GDP and planned aggregate spending implies. For example, if real GDP is 2,500, planned aggregate spending is only 2,300. 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 2,000 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 2,000 implies that planned aggregate spending is greater than real GDP. For example, when real GDP is 1,000, planned aggregate spending is much larger, at 1,400. 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 2,000 implies that firms have underestimated sales, leading to a negative level of unplanned inventory investment in the economy.

By putting together Equations 26-6, 26-1, and 26-4, we can summarize the general relationships among real GDP, planned aggregate spending, 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.

Firms can adjust output and inventory to correct for sales projections that were too high or low, helping to move the economy back to equilibrium.

Monty Rakusen/Getty Images

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 spending.

Staying with our example, if real GDP is 1,000, negative unplanned inventory investment will lead firms to increase production, leading to a rise in real GDP. In fact, this will happen whenever real GDP is less than 2,000—that is, whenever real GDP is less than planned aggregate spending. Conversely, if real GDP is 2,500, positive unplanned inventory investment will lead firms to reduce production, leading to a fall in real GDP. This will happen whenever real GDP is greater than planned aggregate spending.

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 spending in the current period, an outcome known as income–expenditure equilibrium. In Table 26-2, income–expenditure equilibrium is achieved when real GDP is 2,000, 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 GDP.

Figure 26-2 illustrates the concept of income–expenditure equilibrium graphically. Real GDP is on the horizontal axis and planned aggregate spending, AE_Planned, is on the vertical axis. There are two lines in the figure. The solid line is the planned aggregate spending 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).

Income–expenditure equilibrium occurs at E, the point where the planned aggregate spending line, AE_Planned, crosses the 45-degree line. At E, the economy produces real GDP of $2,000 billion per year, the only point at which real GDP equals planned aggregate spending, AE_Planned, and unplanned inventory investment, I_Unplanned, is zero. This is the level of income–expenditure equilibrium GDP, Y*. At any level of real GDP less than Y*, AE_Planned exceeds real GDP. As a result, unplanned inventory investment, I_Unplanned, is negative and firms respond by increasing production. At any level of real GDP greater than Y*, real GDP exceeds AE_Planned. Unplanned inventory investment, I_Unplanned, is positive and firms respond by reducing production.

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 spending is equal to real GDP. 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 spending line. So the point of income–expenditure equilibrium is at E, where the two lines cross. And the income–expenditure equilibrium GDP, Y*, is 2,000—the same outcome we derived in Table 26-2.

Now consider what happens if the economy isn’t in income–expenditure equilibrium. We can see from Figure 26-2 that whenever real GDP is less than Y*, the planned aggregate spending 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 1,000, I_Unplanned is −400. As a consequence, 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 2,500, I_Unplanned is 200. The unanticipated accumulation of inventory leads to a fall in real GDP.

The type of diagram shown in Figure 26-2, which identifies income–expenditure equilibrium as the point at which the planned aggregate spending 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.

The Multiplier Process and Inventory Adjustment

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 in 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 spending 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 spending, let’s turn to understanding what happens when there is a shift of the planned aggregate spending 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 spending?

In our simple model there are only two possible sources of a shift of the planned aggregate spending 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, A) can occur because of a change in aggregate wealth—say, due to a rise in house prices.

When the planned aggregate spending line shifts—when there is a change in the level of planned aggregate spending at any given level of real GDP—there is an autonomous change in planned aggregate spending. Recall that an autonomous change in planned aggregate spending 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 spending affect real GDP in income–expenditure equilibrium?

Table 26-3 and Figure 26-3 start from the same numerical example we used in Table 26-2 and Figure 26-2. They also show the effect of an autonomous increase in planned aggregate spending of 400—what happens when planned aggregate spending is 400 higher at each level of real GDP. Look first at Table 26-3. Before the autonomous increase in planned aggregate spending, the level of real GDP at which planned aggregate spending is equal to real GDP, Y*, is 2,000. After the autonomous change, Y* has risen to 3,000.

This figure illustrates the change in Y* caused by an autonomous increase in planned aggregate spending. The economy is initially at equilibrium point E₁ with an income–expenditure equilibrium GDP, , equal to 2,000. An autonomous increase in AE_Planned of 400 shifts the planned aggregate spending line upward by 400. The economy is no longer in income–expenditure equilibrium: real GDP is equal to 2,000 but AE_Planned is now 2,400, represented by point X. The vertical distance between the two planned aggregate spending lines, equal to 400, represents I_Unplanned = −400—the negative inventory investment that the economy now experiences. Firms respond by increasing production, and the economy eventually reaches a new income–expenditure equilibrium at E₂ with a higher level of income–expenditure equilibrium GDP, , equal to 3,000.

The same result is visible in Figure 26-3. The initial income–expenditure equilibrium is at E₁, where is 2,000. The autonomous rise in planned aggregate spending shifts the planned aggregate spending line up, leading to a new income–expenditure equilibrium at E₂, where is 3,000.

The fact that the rise in income–expenditure equilibrium GDP, from 2,000 to 3,000, is much larger than the autonomous increase in aggregate spending, 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 spending of 400 leads to an increase in Y* from 2,000 to 3,000, a rise of 1,000. So the multiplier in this example is 1,000/400 = 2.5.

We can examine in detail what underlies the multistage multiplier process by looking more closely at Figure 26-3. First, starting from E₁, the autonomous increase in planned aggregate spending leads to a gap between planned aggregate spending and real GDP. This is represented by the vertical distance between X, at 2,400, and E₁, at 2,000. 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 spending will lead to a chain reaction of reductions in real GDP and consumer spending.

We can summarize these results in an equation, where ΔAAE_Planned represents the autonomous change in AE_Planned, and , the subsequent change in income–expenditure equilibrium GDP:

Recalling that the multiplier, 1/(1 − MPC), is greater than 1, Equation 26-7 tells us that the change in income–expenditure equilibrium GDP, ΔY*, is several times as large as the autonomous change in planned aggregate spending, ΔAAE_Planned. 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 ThriftLet’s now consider what happens when there is a slump in consumer spending or investment spending, or both. 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. Economists refer to this phenomenon as the paradox of thrift.

In the paradox of thrift, households and firms cut their spending in anticipation of future tough economic times. 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 for hard times) is “bad” (because it can make everyone worse off).

We’ve seen that 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 spending. Likewise, we know that consumption spending can change as a result of an increase or decrease in the aggregate wealth or expected future disposable income of individuals. But regardless of the source, there are multiplier effects in the economy that magnify the size of the initial change in aggregate spending.

INVENTORIES AND THE END OF A RECESSION

A very clear example of the role of inventories in the multiplier process took place in late 2001, as that year’s recession came to an end.

The driving force behind the recession was a slump in business investment spending. It took several years before investment spending bounced back in the form of a housing boom. Still, the economy did start to recover in late 2001, largely because of an increase in consumer spending—especially on durable goods such as automobiles.

Initially, this increase in consumer spending caught manufacturers by surprise. Figure 26-4 shows changes in real GDP, real consumer spending, and real inventories in each quarter of 2001 and 2002. Notice the surge in consumer spending in the fourth quarter of 2001. It didn’t lead to a lot of GDP growth because it was offset by a plunge in inventories. But in the first quarter of 2002 producers greatly increased their production, leading to a jump in real GDP.

Source: Bureau of Economic Analysis.

Module 26 Review

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