7.1 The Expenditure-Output Model

The Axes of the Expenditure-Output Diagram

The expenditure-output model, sometimes also called the Keynesian cross diagram, determines the equilibrium level of real GDP by the point where the total or aggregate expenditures in the economy are equal to the amount of output produced. The axes of the Keynesian cross diagram presented in Figure 7.3 show real GDP on the horizontal axis as a measure of output and aggregate expenditures on the vertical axis as a measure of spending. 

Remember that GDP can be thought of in several equivalent ways: it measures both the value of spending on final goods and also the value of the production of final goods. All sales of the final goods and services that make up GDP will eventually end up as income for workers, for managers, and for investors and owners of firms. The sum of all the income received for contributing resources to GDP is called national income (Y). At some points in the discussion that follows, it will be useful to refer to real GDP as “national income.” Both axes are measured in real (inflation-adjusted) terms.

The Potential GDP Line and the 45-degree Line

The Keynesian cross diagram contains two lines that serve as conceptual guideposts to orient the discussion. The first is a vertical line showing the level of potential GDP. Potential GDP refers to the quantity of output that the economy can produce with full employment of its labor and physical capital.

The second conceptual line on the Keynesian cross diagram is the 45-degree line, which starts at the origin and reaches up and to the right. A line that stretches up at a 45-degree angle represents the set of points (1, 1), (2, 2), (3, 3) and so on, where the measurement on the vertical axis is equal to the measurement on the horizontal axis. In this diagram, the 45-degree line shows the set of points where the level of aggregate expenditure in the economy, measured on the vertical axis, is equal to the level of output or national income in the economy, measured by GDP on the horizontal axis.

When the macroeconomy is in equilibrium, it must be true that the aggregate expenditures in the economy are equal to the real GDP—because by definition, GDP is the measure of what is spent on final sales of goods and services in the economy. Thus, the equilibrium calculated with a Keynesian cross diagram will always end up where aggregate expenditure and output are equal—which will only occur along the 45-degree line.

The Aggregate Expenditure Schedule

The final ingredient of the Keynesian cross or expenditure-output diagram is the aggregate expenditure schedule, which will show the total expenditures in the economy for each level of real GDP. The intersection of the aggregate expenditure line with the 45-degree line—at point E₀ in Figure 7.3—will show the equilibrium for the economy, because it is the point where aggregate expenditure is equal to output or real GDP. After developing an understanding of what the aggregate expenditures schedule means, we will return to this equilibrium and how to interpret it.

Building the Aggregate Expenditure Schedule

Aggregate expenditure is the key to the expenditure-income model. The aggregate expenditure schedule shows, either in the form of a table or a graph, how aggregate expenditures in the economy rise as real GDP or national income rises. Thus, in thinking about the components of the aggregate expenditure line—consumption, investment, government spending, exports and imports—the key question is how expenditures in each category will adjust as national income rises.

Consumption as a Function of National Income

How do consumption expenditures increase as national income rises? People can do two things with their income: consume it or save it. We will be focusing on disposable income (Y_d) that is after taxes have been paid. The marginal propensity to consume (MPC), is the share of the additional dollar of income a person decides to devote to consumption expenditures. The marginal propensity to save (MPS) is the share of the additional dollar a person decides to save. It must always hold true that:

For example, if the marginal propensity to consume out of the marginal amount of income earned is 0.9, then the marginal propensity to save is 0.1.

With this relationship in mind, consider the relationship among income, consumption, and savings shown in Figure 7.4. (Note that we use “Aggregate Expenditure” on the vertical axis in this and the following figures, because all consumption expenditures are parts of aggregate expenditures.)

An assumption commonly made in this model is that even if income were zero, people would have to consume something. In this example, consumption would be $600 even if income were zero. That portion of spending not tied directly to income is called autonomous consumption. The marginal propensity of consumption and savings tell us how additional income is used. For the graph below, the MPC is 0.8 and the MPS is 0.2. Thus, when income increases by $1,000, consumption rises by $800 and savings rises by $200. At an income of $4,000, total consumption will be the $600 that would be consumed even without any income, plus $4,000 multiplied by the marginal propensity to consume of 0.8, or $ 3,200, for a total of $ 3,800. The total amount of consumption and saving must always add up to the total amount of income. This relationship between income and consumption, illustrated in Figure 7.4 and Table 7.1, is called the consumption function.  The autonomous consumption represents the Y intercept when plotting a consumption function and the MPC becomes the slope of the line, the change in Expenditure divided by the change in Income.

The consumption function can be written as:  C = Autonomous Consumption + MPC (Income level) = A + MPC x Y_d

C = 600 + .8 (2000)

C = 2200

The pattern of consumption shown in Table 7.1 is plotted in Figure 7.4. To calculate consumption, multiply the income level by 0.8, for the marginal propensity to consume, and add $600 autonomous consumption, for the amount that would be consumed even if income was zero. Consumption plus savings must be equal to income.

There are a number of factors other than income that can cause the entire consumption function to shift. Patterns of consumption were presented in section 3.1 of the text where the four components of GDP were first introduced.

When the consumption function moves, it can shift in two ways: either the entire consumption function can move up or down in a parallel manner, this would represent a change in autonomous consumption. Or the slope of the consumption function can shift so that it becomes steeper or flatter. For example, if a tax cut leads consumers to spend more, but does not affect their marginal propensity to consume, it would cause an upward shift to a new consumption function that is parallel to the original one. However, a change in household preferences for saving that reduced the marginal propensity to save would cause the slope of the consumption function to become steeper: that is, if the savings rate is lower, then every increase in income leads to a larger rise in consumption.

Investment as a Function of National Income

Investment decisions are forward-looking, based on expected rates of return. Precisely because investment decisions depend primarily on perceptions about future economic conditions, they do not depend primarily on the level of GDP in the current year. Thus, on a Keynesian cross diagram, the investment function can be drawn as a horizontal line, at a fixed level of expenditure. Figure 7.5 shows an investment function where the level of investment is set at the specific level of 500. Just as a consumption function shows the relationship between consumption levels and real GDP (or national income), the investment function shows the relationship between investment levels and real GDP.

The appearance of the investment function as a horizontal line does not mean that the level of investment never moves. It means only that in the context of this two-dimensional diagram, the level of investment on the vertical aggregate expenditure axis does not vary according to the current level of real GDP on the horizontal axis. However, all the other factors that vary investment—new technological opportunities, expectations about near-term economic growth, interest rates, the price of key inputs, and tax incentives for investment—can cause the horizontal investment function to shift up or down.

Government Spending as a Function of National Income

In the Keynesian cross diagram, government spending appears as a horizontal line, as in Figure 7.6, where government spending is set at a level of 1,300. As in the case of investment spending, this horizontal line does not mean that government spending is unchanging. It means only that government spending changes when Congress decides on a change in the budget, rather than shifting in a predictable way with the current size of the real GDP shown on the horizontal axis.

For the purposes of constructing the basic Keynesian cross diagram it is helpful to make the assumption of disposable income and not include tax calculations in the aggregate expenditure equation.

Exports as a Function of National Income

The export function, which shows how exports change with the level of a country’s own real GDP, is drawn as a horizontal line, as in the example in Figure 7.7 (a) where exports are drawn at a level of $840. Again, as in the case of investment spending and government spending, drawing the export function as horizontal does not imply that exports never change. It just means that they do not change because of what is on the horizontal axis—that is, a country’s own level of domestic production—and instead are shaped by the level of aggregate demand in other countries. More demand for exports from other countries would cause the export function to shift up; less demand for exports from other countries would cause it to shift down.

USING AN ALGEBRAIC APPROACH TO THE EXPENDITURE-OUTPUT MODEL

In the expenditure-output or Keynesian cross model, the equilibrium occurs where the aggregate expenditure line (AE line) crosses the 45-degree line. Given algebraic equations for two lines, the point where they cross can be readily calculated. For this example we will assume a closed economy with no exports or imports. Imagine an economy with the following characteristics.

Y = Real GDP or national income

C = Consumption = 100 + 0.8 Y_d

I = Investment = 400

G = Government spending = 800

Step 1. Determine the aggregate expenditure function. In this case, it is:

AE = C + I + G

AE = 100 + 0.8 Y_d + 400 + 800

Step 2. The equation for the 45-degree line is the set of points where GDP or national income on the horizontal axis is equal to aggregate expenditure on the vertical axis. Thus, the equation for the 45-degree line is: AE = Y.

 

Step 3. The next step is to solve these two equations for Y (or AE, since they will be equal to each other). Substitute Y for AE, producing an equation with only one variable, Y.

Y = AE = 100 + 0.8 Y_d + 400 + 800

Step 4. Work through the algebra and solve for Y.

Y = AE = 100 + 0.8 Y_d + 400 + 800

Y = 1300 + 0.8 Y_d

Y – 0.8 Y_d = 1300

0.2 Y = 1300

Y = 1300/0.2 = 6,500

This algebraic framework is flexible and useful in predicting how economic events and policy actions will affect real GDP.

Step 5. Assume there is a surge of business confidence and investment rises to 500. We can use the aggregate expenditure function to calculate the new equilibrium output.

Y = 100 + 0.8 Y_d + 500 + 800

Y = 1400 + 0.8Y

Y – 0.8Y = 1400

0.2Y = 1400

Y = 7000

 

For issues of policy, the key questions would be how to adjust government spending levels so that the equilibrium level of output is the full employment level. In this case, let the economic parameters be:

Y = National income

C = Consumption = 100 + 0.8 Y_d

I = Investment = 600

G = Government spending = 1,000

Step 6. Calculate the equilibrium for this economy (remember Y = AE).

Y = 100 + 0.8 Y_d + 600 + 1000

             Y – 0.8Y = 1700

0.2Y = 1700

Y = 8500

Step 7. Assume that the full employment level of output is 9,000. What level of government spending would be necessary to reach that level? To answer this question, plug in 9,000 as equal to Y, but leave G as a variable, and solve for G. Thus:

9000 = 100 + 0.8(9000) + 600 + G

9000 -7900 = G

1100 = G

Step 8. Solving this problem arithmetically. The answer is: G = 1,100. In other words, increasing government spending by 100, from its original level of 1,000, to 1,100, would raise output to the full employment level of GDP.

Indeed, the question of how much to increase government spending so that equilibrium output will rise from 8,500 to 9,000 can be answered without working through the algebra, just by using the simple multiplier formula. The multiplier equation in this case is:

Multiplier = 1 / (1 – MPC) = 1 /(1-0.8) = 1/.2 = 5

Thus, to raise output by 500 would require an increase in government spending of 500/5=100, which is the same as the answer derived from the algebraic calculation.

Building the Combined Aggregate Expenditure Function

The aggregate expenditure function is formed by stacking on top of each other the consumption function (after taxes), the investment function, the government spending function, and the export function. The point at which the aggregate expenditure function intersects the vertical axis will be determined by the levels of investment, government, and export expenditures—which do not vary with national income. The upward slope of the aggregate expenditure function will be determined by the marginal propensity consume. A higher marginal propensity to consume will make the slope of the aggregate expenditure function steeper—because out of any extra income, more is being spent goods and services.

The equilibrium occurs where national income is equal to aggregate expenditure, which is shown on the graph as the point where the aggregate expenditure schedule crosses the 45-degree line. In this example, the equilibrium occurs at 6,000. This equilibrium is the level of national income where aggregate expenditure is equal to national income.

Equilibrium in the Keynesian Cross Model

With the aggregate expenditure line in place, the next step is to relate it to the two other elements of the Keynesian cross diagram. Thus, the first subsection interprets the intersection of the aggregate expenditure function and the 45-degree line, while the next subsection relates this point of intersection to the potential GDP line.

Where Equilibrium Occurs

The point where the aggregate expenditure line that is constructed from C + I + G + X – M crosses the 45-degree line will be the equilibrium for the economy. It is the only point on the aggregate expenditure line where the total amount being spent on aggregate demand equals the total level of production. In Figure 7.6, this point of equilibrium (E₀) happens at 6,000.

The meaning of “equilibrium” remains the same; that is, equilibrium is a point of balance where no incentive exists to shift away from that outcome. To understand why the point of intersection between the aggregate expenditure function and the 45-degree line is a macroeconomic equilibrium, consider what would happen if an economy found itself to the right of the equilibrium point E, say point H in Figure 7.7, where output is higher than the equilibrium. At point H, the level of aggregate expenditure is below the 45-degree line, so that the level of aggregate expenditure in the economy is less than the level of output. As a result, at point H, output is piling up unsold—not a sustainable state of affairs.

Conversely, consider the situation where the level of output is at point L—where real output is lower than the equilibrium. In that case, the level of aggregate demand in the economy is above the 45-degree line, indicating that the level of aggregate expenditure in the economy is greater than the level of output. When the level of aggregate demand has emptied the store shelves, it cannot be sustained, either. Firms will respond by increasing their level of production. Thus, the equilibrium must be the point where the amount produced and the amount spent are in balance, at the intersection of the aggregate expenditure function and the 45-degree line.

FINDING EQUILIBRIUM

Table 7.2 gives some information on an economy. The Keynesian model assumes that there is some level of consumption even without income. That amount is $236 – $216 = $20. $20 will be consumed when national income equals zero. Let us assume the MPC = 0.9. The level of investment is $70, the level of government spending is $80, and assume there are no exports or imports. Given these values, you need to complete Table 7.2  and then answer these questions:

• What is the consumption function?
• What is the equilibrium?
• Why is a national income of $1,300 not at equilibrium?
• How do expenditures and output compare at this point?

Table 7.2 National Income

National Income	Consumption	Investment	Government Spending	Aggregate Expenditures
$1300
$1500
$1700
$1900
$2100

Step 1. Calculate the amount consumption for each level of national income using the consumption function:

C = $20 + 0.9 Y

= $20 + 0.9 ($1300)

= $1190

Step 2. Fill in the amounts for Investment Spending (I) and Government Spending (G). Remember that these do not change as national income changes.

Step 3. Find aggregate expenditure by adding C + I + G for each level of national income. Your completed table should look like Table 7.3.

Table 7.3 Aggregate Expenditures

National Income	Consumption	Investment	Government Spending	Aggregate Expenditures
$1300	$1190	$70	$80	$1340
$1500	$1370	$70	$80	$1520
$1700	$1550	$70	$80	$1700
$1900	$1730	$70	$80	$1880
$2100	$1910	$70	$80	$2060

Step 4. Answer the question: What is equilibrium? Equilibrium occurs where AE = Y. Table 7.3 shows that equilibrium occurs where national income equals aggregate expenditure at $1700.

Step 5. Find equilibrium mathematically, knowing that national income is equal to aggregate expenditure.

Y = AE

    = C + I + G

    = $20 + 0.9 Y+ $70 + $80

    = $170 + 0.9 Y

 

Solve for Y.

                  Y = $170 + 0.9Y

Y – 0.9 Y = $170

0.1 Y = $170

       Y = $170/0.1 = $1700

Step 6. Answer this question: Why is a national income of $1300 not an equilibrium? At national income of $1300, aggregate expenditures are $1340.

Step 7. Answer this question: How do expenditures and output compare at this point? Aggregate expenditures cannot exceed output (GDP) in the long run, since there would not be enough goods to be bought. This would represent an inflationary gap.

Recessionary and Inflationary Gaps

In the Keynesian cross diagram, if the aggregate expenditure line intersects the 45-degree line at the level of potential GDP, then the economy is in sound shape. There is no recession, and unemployment is low. But there is no guarantee that the equilibrium will occur at the potential GDP level of output. The equilibrium might be higher or lower.

For example, Figure 7.10 (a) illustrates a situation where the aggregate expenditure line intersects the 45-degree line at point E₀, which is a real GDP of $6,000, and which is below the potential GDP of $7,000. In this situation, the level of aggregate expenditure is too low for GDP to reach its full employment level, and unemployment will occur. The distance between an output level like E₀ that is below potential GDP and the level of potential GDP is called a recessionary gap. Because the equilibrium level of real GDP is so low, firms will not wish to hire the full employment number of workers, and unemployment will be high.

What might cause a recessionary gap? Anything that shifts the aggregate expenditure line down is a potential cause of recession, including a decline in consumption, a rise in savings, a fall in investment, a drop in government spending or a rise in taxes, or a fall in exports or a rise in imports. Moreover, an economy that is at equilibrium with a recessionary gap may just stay there and suffer high unemployment for a long time; remember, the meaning of equilibrium is that there is no particular adjustment of prices or quantities in the economy to chase the recession away.

The appropriate response to a recessionary gap is for the government to reduce taxes or increase spending so that the aggregate expenditure function shifts up from AE₀ to AE₁. When this shift occurs, the new equilibrium E₁ now occurs at potential GDP as shown in Figure 7.10 (a).

Conversely, Figure 7.10 (b) shows a situation where the aggregate expenditure schedule (AE₀) intersects the 45-degree line above potential GDP. The gap between the level of real GDP at the equilibrium E₀ and potential GDP is called an inflationary gap. The inflationary gap also requires a bit of interpreting. After all, a naïve reading of the Keynesian cross diagram might suggest that if the aggregate expenditure function is just pushed up high enough, real GDP can be as large as desired—even doubling or tripling the potential GDP level of the economy. This implication is clearly wrong. An economy faces some supply-side limits on how much it can produce at a given time with its existing quantities of workers, physical and human capital, technology, and market institutions.

The inflationary gap should be interpreted, not as a literal prediction of how large real GDP will be, but as a statement of how much extra aggregate expenditure is in the economy beyond what is needed to reach potential GDP. An inflationary gap suggests that because the economy cannot produce enough goods and services to absorb this level of aggregate expenditures, the spending will instead cause an inflationary increase in the price level. In this way, even though changes in the price level do not appear explicitly in the Keynesian cross equation, the notion of inflation is implicit in the concept of the inflationary gap.

The appropriate Keynesian response to an inflationary gap is shown in Figure 7.10 (b). The original intersection of aggregate expenditure line AE₀ and the 45-degree line occurs at $8,000, which is above the level of potential GDP at $7,000. If AE₀ shifts down to AE₁, so that the new equilibrium is at E₁, then the economy will be at potential GDP without pressures for inflationary price increases. The government can achieve a downward shift in aggregate expenditure by increasing taxes on consumers or firms, or by reducing government expenditures.

The Multiplier Effect

The Keynesian policy prescription has one final twist. Assume that for a certain economy, the intersection of the aggregate expenditure function and the 45-degree line is at a GDP of 700, while the level of potential GDP for this economy is $800. By how much does government spending need to be increased so that the economy reaches the full employment GDP? The obvious answer might seem to be $800 – $700 = $100; so raise government spending by $100. But that answer is incorrect. A change of, for example, $100 in government expenditures will have an effect of more than $100 on the equilibrium level of real GDP. The reason is that a change in aggregate expenditures circles through the economy: households buy from firms, firms pay workers and suppliers, workers and suppliers buy goods from other firms, those firms pay their workers and suppliers, and so on. In this way, the original change in aggregate expenditures is actually spent more than once. This is called the multiplier effect: An initial increase in spending, cycles repeatedly through the economy and has a larger impact than the initial dollar amount spent.

Calculating the Multiplier

Fortunately for everyone who is not carrying around a computer with a spreadsheet program to project the impact of an original increase in expenditures over 20, 50, or 100 rounds of spending, there is a formula for calculating the multiplier.

If the MPC is equal to 1 – MPS, or 0.7. The spending multiplier would be:

A change in spending of $100 multiplied by the spending multiplier of 3.33 is equal to a change in GDP of $333.  The size of the multiplier is determined by what proportion of the marginal dollar of income goes into actual consumption. An MPC of 0.9 will result in a larger multiplier of 1/0.1 = 10. An MPC of 0.8 will result in a multiplier of 1/.02 = 5.

Calculating Keynesian Policy Interventions

Returning to the original question: How much should government spending be increased to produce a total increase in real GDP of $100? If the goal is to increase aggregate demand by $100, and the multiplier is 3.33, then the increase in government spending to achieve that goal would be $100/3.33 = $30. Government spending of approximately $30, when combined with a multiplier of 3.33, produces an overall increase in real GDP of $100, restoring the economy to potential GDP of $800, as Figure 7.11 shows.

The multiplier effect is also visible on the Keynesian cross diagram. Figure 7.11 shows the example we have been discussing: a recessionary gap with an equilibrium of $700, potential GDP of $800, the slope of the aggregate expenditure function (AE₀) determined by the assumptions that the marginal propensity to consume is 0.7. At AE₁, the aggregate expenditure function is moved up to reach potential GDP.

Now, compare the vertical shift upward in the aggregate expenditure function, which is $30, with the horizontal shift outward in real GDP, which is $100 (as these numbers were calculated earlier). The rise in real GDP is more than double the rise in the aggregate expenditure function. (Similarly, if you look back at Figure 7.10, you will see that the vertical movements in the aggregate expenditure functions are smaller than the change in equilibrium output that is produced on the horizontal axis. Again, this is the multiplier effect at work.) In this way, the power of the multiplier is apparent in the income–expenditure graph, as well as in the arithmetic calculation.

The multiplier does not just affect government spending, but applies to any change in the economy. Say that business confidence declines and investment falls off, or that the economy of a leading trading partner slows down so that export sales decline. These changes will reduce aggregate expenditures, and then will have an even larger effect on real GDP because of the multiplier effect.

Multiplier Tradeoffs: Stability versus the Power of Macroeconomic Policy

Is an economy healthier with a high multiplier or a low one? With a high multiplier, any change in aggregate demand will tend to be substantially magnified, and so the economy will be more unstable. With a low multiplier, by contrast, changes in aggregate demand will not be multiplied much, so the economy will tend to be more stable.

However, with a low multiplier, government policy changes in taxes or spending will tend to have less impact on the equilibrium level of real output. With a higher multiplier, government policies to raise or reduce aggregate expenditures will have a larger effect. Thus, a low multiplier means a more stable economy, but also weaker government macroeconomic policy, while a high multiplier means a more volatile economy, but also an economy in which government macroeconomic policy is more powerful.