Learning Objectives

By the end of this section, you will be able to:

Explain what the expenditure-output model/Keynesian cross diagram shows and what the equilibrium point on the diagram represents.
Analyze the consumption function, investment function, government spending, taxes, imports and exports is relation to the expenditure-output model.
Analyze the location of an equilibrium point on an expenditure-output model to determine characteristics of the economy, including the presence of a recessionary gap or an inflationary gap.
Evaluate the stability of an economy based on the multiplier effect in relation to the expenditure-output model.

The fundamental ideas of Keynesian economics were developed before the AD/AS model was popularized. From the 1930s until the 1970s, Keynesian economics was usually explained with a different model, known as the expenditure–output approach. This approach is rooted strongly in the fundamental assumptions of Keynesian economics: It focuses on the total amount of spending in the economy, with no explicit mention of aggregate supply or of the price level, although, as you will see, it is possible to draw some inferences about aggregate supply and price levels based on the diagram.

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 11.7 show real GDP on the horizontal axis as a measure of output, and aggregate expenditures on the vertical axis as a measure of spending.

A graph is shown plotting Aggregate Expenditure on the y-axis and Real DGP on the x-axis. A line extends from the origin at 45 degrees. On this line is a point E sub zero, with a dashed line extending to the y-axis to the left. The label along the y-axis is A E sub zero. Another dashed line extends from E sub zero downward to the x-axis, and the label along the x-axis is Y sub zero. Running parallel to this second dashed line is a solid line slightly to the right extending upward from the x-axis, labeled

Figure 11.7 The Expenditure-Output Diagram The aggregate expenditure-output model shows aggregate expenditures on the vertical axis and real GDP on the horizontal axis. A vertical line shows potential GDP where full employment occurs. The 45° line shows all points where aggregate expenditures and output are equal. The aggregate expenditure schedule shows how total spending or aggregate expenditure increases as output or real GDP rises. The intersection of the aggregate expenditure schedule and the 45° line is the equilibrium. Equilibrium occurs at E₀, where aggregate expenditure (AE₀) is equal to the output level (Y₀).

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 is 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° Line

The Keynesian cross diagram contains two lines that serve as conceptual guideposts to orient the discussion. The first is a vertical line that shows the level of potential GDP. Potential GDP means the same thing here that it means in the AD/AS diagrams: It refers to the quantity of output 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° line, which starts at the origin and reaches up and to the right. A line that stretches up at a 45° 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° 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 occurs only along the 45° line.

The Aggregate Expenditure Schedule

The final ingredient of the Keynesian cross or expenditure-output diagram is the aggregate expenditure schedule, which shows the total expenditures in the economy for each level of real GDP. The intersection of the aggregate expenditure line with the 45° line—at point E₀ in Figure 11.7—shows 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 return to this equilibrium and examine 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 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—for the moment, let’s ignore the need to pay taxes with some of it. Each person who receives an additional dollar faces this choice. 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

MPC + MPS = 1.

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 11.8. 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 is $600 even if income were zero. Then, 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 is 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. Exactly how a situation of zero income and negative savings works in practice is not important, because even low-income societies are not literally at zero income, so the point is hypothetical. This relationship between income and consumption, illustrated in Figure 11.8 and Table 11.2, is called the consumption function.

A graph is shown plotting Aggregate Expenditure along the y-axis and Real GDP (Y) along the x-axis. The y-axis has tick marks noted at $2000, $4000, $6000, $8000, and $10,000. The x-axis also has tick marks noting these same increments. A line extends from slightly above the origin, upward and to the right, and has dots marking every $1000 on average. This line is labeled Consumption function.

Figure 11.8 The Consumption Function In the expenditure-output model, how does consumption increase with the level of national income? Output on the horizontal axis is conceptually the same as national income because the value of all final output produced and sold must be income to someone, somewhere in the economy. At a national income level of zero, $600 is consumed. Then, each time income rises by $1,000, consumption rises by $800, because, in this example, the marginal propensity to consume is 0.8.

The pattern of consumption shown in Table 11.2 is plotted in Figure 11.8. To calculate consumption, multiply the income level by 0.8, for the marginal propensity to consume, and add $600, for the amount that would be consumed even if income was zero. Consumption plus savings must be equal to income.

Income	Consumption	Savings
$0	$600	–$600
$1,000	$1,400	–$400
$2,000	$2,200	–$200
$3,000	$3,000	$0
$4,000	$3,800	$200
$5,000	$4,600	$400
$6,000	$5,400	$600
$7,000	$6,200	$800
$8,000	$7,000	$1,000
$9,000	$7,800	$1,200

Table 11.2 The Consumption Function

However, a number of factors other than income can also cause the entire consumption function to shift. These factors are summarized in the earlier discussion of consumption and are listed in Table 11.2. When the consumption function moves, it can shift in two ways: The entire consumption function can move up or down in a parallel manner, or the slope of the consumption function can shift so 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 causes an upward shift to a new consumption function is parallel to the original one. However, a change in household preferences for saving that reduces the marginal propensity to save causes 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 11.9 shows an investment function where the level of investment is, for the sake of concreteness, 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.

Figure 11.9 The Investment Function The investment function is drawn as a flat line because investment is based on interest rates and expectations about the future, and so it does not change with the level of current national income. In this example, investment expenditures are at a level of 500. However, changes in factors such as technological opportunities, expectations about near-term economic growth, and interest rates would all cause the investment function to shift upward or downward.

The appearance of the investment function as a horizontal line does not mean 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 upward or downward.

Government Spending and Taxes as a Function of National Income

In the Keynesian cross diagram, government spending appears as a horizontal line, as in Figure 11.10, where government spending is set at a level of $1,300. As in the case of investment spending, this horizontal line does not mean 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.

Figure 11.10 The Government Spending Function The level of government spending is determined by political factors, not by the level of real GDP in a given year. Thus, government spending is drawn as a horizontal line. In this example, government spending is at a level of $1,300. Congressional decisions to increase government spending cause this horizontal line to shift upward whereas decisions to reduce spending would cause it to shift downward.

The situation of taxes is different because taxes often rise or fall with the volume of economic activity. For example, income taxes are based on the level of income earned, and sales taxes are based on the amount of sales made, and both income and sales tend to be higher when the economy is growing and lower when the economy is in a recession. For the purposes of constructing the basic Keynesian cross diagram, it is helpful to view taxes as a proportionate share of GDP. In the United States, for example, taking federal, state, and local taxes together, the government typically collects about 30 percent to 35 percent of income as taxes.

Table 11.3 revises the earlier table on the consumption function so that it takes taxes into account. The first column shows national income. The second column calculates taxes, which in this example are set at a rate of 30 percent, or 0.3. The third column shows after-tax income; that is, total income minus taxes. The fourth column then calculates consumption in the same manner as before: Multiply after-tax income by 0.8, representing the marginal propensity to consume, and then add $600, for the amount that would be consumed even if income were zero. When taxes are included, the marginal propensity to consume is reduced by the amount of the tax rate, so each additional dollar of income results in a smaller increase in consumption than before taxes. For this reason, the consumption function, with taxes included, is flatter than the consumption function without taxes, as Figure 11.11 shows.

Figure 11.11 The Consumption Function before and after Taxes The upper line repeats the consumption function from Figure 11.8. The lower line shows the consumption function if taxes must first be paid on income, and then consumption is based on after-tax income.

Income	Taxes	After-Tax Income	Consumption	Savings
$0	$0	$0	$600	–$600
$1,000	$300	$700	$1,160	–$460
$2,000	$600	$1,400	$1,720	–$320
$3,000	$900	$2,100	$2,280	–$180
$4,000	$1,200	$2,800	$2,840	–$40
$5,000	$1,500	$3,500	$3,400	$100
$6,000	$1,800	$4,200	$3,960	$240
$7,000	$2,100	$4,900	$4,520	$380
$8,000	$2,400	$5,600	$5,080	$520
$9,000	$2,700	$6,300	$5,640	$660

Table 11.3 The Consumption Function Before and After Taxes

Exports and Imports 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 11.12 (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 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 causes the export function to shift upward; less demand for exports from other countries causes it to shift downward.

Two graphs are shown plotting Aggregate Expenditure along the y-axis and Real GDP (Y) along the x-axis. The y-axis has tick marks noted at $2000, $4000, $6000, $8000, and $10,000. The x-axis also has tick marks noting these same increments. The left graph, titled The export function, has a horizontal line extending from the y-axis to the right at the value of $840. A horizontal line extends from the y-axis to the right at the value of $1300 and is labeled Government spending. The graph on the right, l

Figure 11.12 The Export and Import Functions (a) The export function is drawn as a horizontal line because exports are determined by the buying power of other countries and thus do not change with the size of the domestic economy. In this example, exports are set at $840. However, exports can shift upward or downward, depending on buying patterns in other countries. (b) The import function is drawn in negative territory because expenditures on imported products are subtracted from expenditures in the domestic economy. In this example, the marginal propensity to import is 0.1, so imports are calculated by multiplying the level of income by –0.1.

Imports are drawn in the Keynesian cross diagram as a downward-sloping line, with the downward slope determined by the marginal propensity to import (MPI), out of national income. In Figure 11.12 (b), the marginal propensity to import is 0.1. Thus, if real GDP is $5,000, imports are $500; if national income is $6,000, imports are $600, and so on. The import function is drawn as downward sloping and negative, because it represents a subtraction from the aggregate expenditures in the domestic economy. A change in the marginal propensity to import, perhaps as a result of changes in preferences, alters the slope of the import function.

Work It Out

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° line. Given algebraic equations for two lines, the point where they cross can be calculated easily. Imagine an economy with the following characteristics.

Y = Real GDP or national income

T = Taxes = 0.3Y

C = Consumption = 140 + 0.9(Y – T)

I = Investment = 400

G = Government spending = 800

X = Exports = 600

M = Imports = 0.15Y

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

\begin{array}{rcl} AE & = & C + I + G + X – M \\ AE & = & 140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y. \end{array}

Step 2. The equation for the 45° 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° 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

\begin{array}{rcl} Y & = & 140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y. \end{array}

Step 4. Insert the term 0.3Y for the tax rate T. This produces an equation with only one variable, Y.

Step 5. Work through the algebra and solve for Y

\begin{array}{rcl} Y & = & 140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.15Y \\ Y & = & 140 + 0.9Y – 0.27Y + 1,800 – 0.15Y \\ Y & = & 1,940 + 0.48Y \\ Y – 0.48Y & = & 1,940 \\ 0.52Y & = & 1,940 \\ \frac{0.52Y}{0.52} & = & \frac{1,940}{0.52} \\ Y & = & 3,730. \end{array}

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

Step 6. Say, for example, that because of changes in the relative prices of domestic and foreign goods, the marginal propensity to import falls to 0.1. Calculate the equilibrium output when the marginal propensity to import is changed to 0.1.

\begin{array}{rcl} Y & = & 140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.1Y \\ Y & = & 1,940 – 0.53Y \\ 0.47Y & = & 1,940 \\ Y & = & 4,127. \end{array}

Step 7. Because of a surge of business confidence, investment rises to 500. Calculate the equilibrium output.

\begin{array}{rcl} Y & = & 140 + 0.9(Y – 0.3Y) + 500 + 800 + 600 – 0.15Y \\ Y & = & 2,040 + 0.48Y \\ Y – 0.48Y & = & 2,040 \\ 0.52Y & = & 2,040 \\ Y & = & 3,923. \end{array}

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

Y = National income

T = Taxes = 0.3Y

C = Consumption = 200 + 0.9(Y – T).

I = Investment = 600

G = Government spending = 1,000

X = Exports = 600

Y = Imports = 0.1(Y – T)

Step 8. Calculate the equilibrium for this economy (remember, Y = AE).

\begin{array}{l} After-tax income $ 240 \\ Imports of 0 .2 or 20 percent of Y - T \underline{× 0.2} \\ Imports $ 48. \end{array}

Step 9. Assume that the full employment level of output is 6,000. What level of government spending is necessary to reach that level? To answer this question, plug in 6,000 as equal to Y, but leave G as a variable, and solve for G. Thus

6,000 = 200 + 0.9[6,000 – 0.3(6,000)] + 600 + G + 600 – 0.1[6,000 – 0.3(6,000)].

Step 10. Solve this problem arithmetically. The answer is G = 1,240. In other words, increasing government spending by 240, from its original level of 1,000, to 1,240, would raise output to the full employment level of GDP.

Indeed, the question of how much to increase government spending so that equilibrium output rises from 5,454 to 6,000 can be answered without working through the algebra. Just use the multiplier formula. The multiplier equation in this case is

\begin{array}{rcl} \frac{1}{1 – 0.56} & = & 2.27. \end{array} \begin{array}{l}  \end{array}

Thus, to raise output by 546 requires an increase in government spending of 546/2.27 = 240, which is the same as the answer derived from the algebraic calculation.

This algebraic framework is highly flexible. For example, taxes can be treated as a total set by political considerations, such as government spending, and not dependent on national income. Imports might be based on before-tax income, not after-tax income. For certain purposes, it may be helpful to analyze the economy without exports, and imports. A more complicated approach could divide up consumption, investment, government, exports and imports into smaller categories, or to build in some variability in the rates of taxes, savings, and imports. A wise economist shapes the model to fit the specific issue under investigation.

Building the Combined Aggregate Expenditure Function

All the components of aggregate demand—consumption, investment, government spending, and the trade balance—are now in place to build the Keynesian cross diagram. Figure 11.13 builds up an aggregate expenditure function, based on the numerical illustrations of C, I, G, X, and M that have been used throughout this text. The first three columns in Table 11.4 are lifted from the earlier Table 11.3, which showed how to bring taxes into the consumption function. The first column is real GDP or national income, which is what appears on the horizontal axis of the income-expenditure diagram. The second column calculates after-tax income, based on the assumption, in this case, that 3 percent of real GDP is collected in taxes. The third column is based on a marginal propensity to consume (MPC) of 0.8, so that as after-tax income rises by $700 from one row to the next, consumption rises by $560 (700 × 0.8) from one row to the next. Investment, government spending, and exports do not change with the level of current national income. In the previous discussion, investment was $500, government spending was $1,300, and exports were $840, for a total of $2,640. This total is shown in the fourth column. Imports are 0.1 of real GDP in this example, and the level of imports is calculated in the fifth column. The final column, aggregate expenditures, expresses C + I + G + X – M. This aggregate expenditure line is illustrated in Figure 11.13.

Figure 11.13 A Keynesian Cross Diagram Each combination of national income and aggregate expenditure—after-tax consumption, government spending, investment, exports, and imports—is graphed. Equilibrium occurs when aggregate expenditure is equal to national income. This occurs where the aggregate expenditure schedule crosses the 45° line, at a real GDP of $6,000. Potential GDP in this example is $7,000, so equilibrium is occurring at a level of output or real GDP below the potential GDP level.

National Income	After-Tax Income	Consumption	Government Spending + Investment + Exports	Imports	Aggregate Expenditure
$3,000	$2,100	$2,280	$2,640	$300	$4,620
$4,000	$2,800	$2,840	$2,640	$400	$5,080
$5,000	$3,500	$3,400	$2,640	$500	$5,540
$6,000	$4,200	$3,960	$2,640	$600	$6,000
$7,000	$4,900	$4,520	$2,640	$700	$6,460
$8,000	$5,600	$5,080	$2,640	$800	$6,920
$9,000	$6,300	$5,640	$2,640	$900	$7,380

Table 11.4 National Income-Aggregate Expenditure Equilibrium

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, the export function, and the import function. The point at which the aggregate expenditure function intersects the vertical axis is 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 is determined by the marginal propensity to save, the tax rate, and the marginal propensity to import. A higher marginal propensity to save, a higher tax rate, and a higher marginal propensity to import all make the slope of the aggregate expenditure function flatter, because out of any extra income, more is going to savings or taxes or imports and less is going to spending on domestic goods and services.

Equilibrium occurs when national income is equal to aggregate expenditure, which is shown on the graph as the point where the aggregate expenditure schedule crosses the 45° line. In this example, equilibrium occurs at $6,000. Equilibrium can also be read off Table 11.4. It 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° line; the next subsection relates this point of intersection to the potential GDP line.

Where Equilibrium Occurs

The point where the aggregate expenditure line, which is constructed from C + I + G + X – M, crosses the 45° line is 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 11.13, this point of equilibrium (E₀) happens at $6,000, which can also be read off Table 11.4.

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° line is a macroeconomic equilibrium, consider what happens if an economy finds itself to the right of the equilibrium point E—say, at point H in Figure 11.14—where output is greater than the equilibrium. At point H, the level of aggregate expenditure is below the 45° 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.

A graph is shown plotting Aggregate Expenditure on the y-axis and Real GDP on the x-axis. A line extends from the origin at a 45 degree angle. At point L, output is less than aggregate expenditure. At point H, output is greater than aggregate expenditure.

Figure 11.14 Equilibrium in the Keynesian Cross Diagram If output is above the equilibrium level, at H, then the real output is greater than the aggregate expenditure in the economy. This pattern cannot hold, because it would means goods are produced, but are piling up unsold. If output is below the equilibrium level, at L, then aggregate expenditure is greater than output. This pattern cannot hold either, because it means spending exceeds the number of goods being produced. Only point E can be at equilibrium, where output, or national income and aggregate expenditure, are equal. Equilibrium (E) must lie on the 45° line, which is the set of points where national income and aggregate expenditure are equal.

Conversely, consider the situation when the level of output is at point L, where real output is less than equilibrium. In this case, the level of aggregate demand in the economy is above the 45° line, indicating 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. Firms respond by increasing their level of production. Thus, 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° line.

Work It Out

Finding Equilibrium

Table 11.5 gives some information on an economy. The Keynesian model assumes there is some level of consumption even without income. In this example, this amount is $236 – $216 = $20; $20 is consumed when national income equals zero. Assume that taxes are 0.2 of real GDP. Let the marginal propensity to save of after-tax income be 0.1. The level of investment is $70, the level of government spending is $80, and the level of exports is $50. Imports are 0.2 of after-tax income.

Given these values, complete Table 11.5 and then answer these questions:

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

National Income	Taxes	After-Tax income	Consumption	I + G + X	Imports	Aggregate Expenditures
$300			$236
$400
$500
$600
$700

Table 11.5

Step 1. Calculate the amount of taxes for each level of national income—reminder—GDP equals national income for each level of national income using the following as an example

\begin{array}{l} National income (Y)                          $300 \\ Taxes = 0 .2 or 20% \underline{× 0 .2} \\ Tax amount (T)                                    $60. \end{array}

Step 2. Calculate after-tax income by subtracting the tax amount from national income for each level of national income using the following as an example

\begin{array}{l} National income minus taxes       $300 \\ \underline{$60} \\ After-tax income                           $60. \end{array}

Step 3. Calculate consumption. The marginal propensity to save is given as 0.1. This means that the marginal propensity to consume is 0.9, since MPS + MPC = 1. Therefore, multiply 0.9 by the after-tax income amount using the following as an example

\begin{array}{l} After-tax income     $240 \\ MPC \underline{× 0 .9} \\ Consumption           $216. \end{array}

Step 4. Consider why the table shows a consumption of $236 in the first row. As mentioned earlier, the Keynesian model assumes there is some level of consumption even without income. That amount is $236 – $216 = $20.

Step 5. There is now enough information to write the consumption function. The consumption function is found by figuring out the level of consumption that happens when income is zero.

Remember that

C = Consumption when national income is zero + MPC (after-tax income).

Let C represent the consumption function, Y represent national income, and T represent taxes.

\begin{array}{rcl} C & = & $20 + 0.9(Y – T) \\ = & $20 + 0.9($300 – $60) \\ = & $236. \end{array}

Step 6. Use the consumption function to find consumption at each level of national income.

Step 7. Add investment (I), government spending (G), and exports (X). Remember, these do not change as national income changes

Step 8. Find imports, which are 0.2 of after-tax income at each level of national income. For example,

\begin{array}{l} After-tax income                                $240 \\ Imports of 0 .2 or 20% of Y – T \underline{× 0 .2} \\ Imports                                                $48 . \end{array}

Step 9. Find aggregate expenditure by adding C + I + G + X – I for each level of national income. Your completed table should look like Table 11.6.

National Income (Y)	Tax = 0.2 × Y(T)	After-Tax income (Y – T)	Consumption C = $20 + 0.9(Y – T)	I + G + X	Minus Imports (M)	Aggregate Expenditures AE = C + I + G + X – M
$300	$60	$240	$236	$200	$48	$388
$400	$80	$320	$308	$200	$64	$444
$500	$100	$400	$380	$200	$80	$500
$600	$120	$480	$452	$200	$96	$556
$700	$140	$560	$524	$200	$112	$612

Table 11.6 Completed Table

Step 10. Answer the question: What is equilibrium? Equilibrium occurs where AE = Y. Table 11.6 shows that equilibrium occurs where national income equals aggregate expenditure at $500.

Step 11. Find equilibrium mathematically, knowing that national income is equal to aggregate expenditure.

\begin{array}{llll} Y & = & AE \\ = & C + I + G + X – M \\ = & $20 + 0.9(Y – T) + $70 + $80 + $50 – 0.2(Y – T) \\ = & $220 + 0.9(Y – T) – 0.2(Y – T). \end{array}

Since T is 0.2 of national income, substitute T with 0.2 Y so that

\begin{array}{llll} Y & = & $220 + 0.9(Y – 0.2Y) – 0.2(Y – 0.2Y) \\ = & $220 + 0.9Y – 0.18Y – 0.2Y + 0.04Y \\ = & $220 + 0.56Y. \end{array}

Solve for Y.

\begin{array}{rcl} Y & = & $220 + 0.56Y \\ Y – 0.56Y & = & $220 \\ 0.44Y & = & $220 \\ \frac{0.44Y}{0.44} & = & \frac{$220}{0.44} \\ Y & = & $500. \end{array}

Step 12. Answer this question: Why is a national income of $300 not an equilibrium? At a national income of $300, aggregate expenditures are $388. The two values are not equal, thus they are not at equilibrium.

Step 13. Answer this question: How do expenditures and output compare at this point? Aggregate expenditures cannot exceed output (GDP) in the long run, because there are not be enough goods to be bought.

Recessionary and Inflationary Gaps

In the Keynesian cross diagram, if the aggregate expenditure line intersects the 45° 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 equilibrium occurs at the potential GDP level of output. Equilibrium might be higher or lower.

For example, Figure 11.15 (a) illustrates a situation in which the aggregate expenditure line intersects the 45° 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 occurs.

The distance between an output level such as E₀, which 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.

Two graphs are shown plotting Aggregate Expenditure in dollars along the y-axis and Real GDP in billions of constant dollars along the x-axis. The y-axis has tick marks noted at $2000, $4000, $6000, $8000, and $10,000. The x-axis also has tick marks noting these same increments. The left graph is a policy to address a recessionary gap, and the right column is a policy to address an inflationary gap.

Figure 11.15 Addressing Recessionary and Inflationary Gaps (a) If equilibrium occurs at an output below potential GDP, then a recessionary gap exists. The policy solution to a recessionary gap is to shift the aggregate expenditure schedule up from AE₀ to AE₁ using policies such as tax cuts or government spending increases. Then, the new equilibrium, E₁, occurs at potential GDP. (b) If equilibrium occurs at an output above potential GDP, then an inflationary gap exists. The policy solution to an inflationary gap is to shift the aggregate expenditure schedule down from AE₀ to AE₁ using policies such as tax increases or spending cuts. Then, the new equilibrium, E₁, occurs at potential GDP.

What might cause a recessionary gap? Anything that shifts the aggregate expenditure line downward 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 the aggregate expenditure function shifts upward from AE₀ to AE₁. When this shift occurs, the new equilibrium, E₁, now occurs at potential GDP, as shown in Figure 11.15 (a).

Conversely, Figure 11.15 (b) shows a situation in which the aggregate expenditure schedule (AE₀) intersects the 45° line above potential GDP. The gap between the level of real GDP at equilibrium E₀ and potential GDP is called an inflationary gap. The inflationary gap also requires a bit of interpreting. After all, a naive 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 instead causes 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 11.15 (b). The original intersection of aggregate expenditure line AE₀ and the 45 line occurs at $8,000, which is above the level of potential GDP at $7,000. If AE₀ shifts downward to AE₁, so that the new equilibrium is at E₁, then the economy is at potential GDP without the 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° line is at a GDP of $700 whereas 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 has 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.

How Does the Multiplier Work?

To understand how the multiplier effect works, return to the example in which the current equilibrium in the Keynesian cross diagram is a real GDP of $700, or $100 short of the $800 needed to be at full employment, potential GDP. If the government spends $100 to close this gap, someone in the economy receives that spending and can treat it as income. Assume that those who receive this income pay 30 percent in taxes, save 10 percent of after-tax income, spend 10 percent of total income on imports, and then spend the rest on domestically produced goods and services.

As shown in the calculations in Figure 11.16 and Table 11.7, of the original $100 in government spending, $53 is left to spend on domestically produced goods and services. That $53, which was spent, becomes income to someone, somewhere in the economy. Those who receive that income also pay 30 percent in taxes, save 10 percent of after-tax income, and spend 10 percent of total income on imports, as shown in Figure 11.16, so that an additional $28.09, that is, 0.53 × $53, is spent in the third round. The people who receive that income then pay taxes, save, and buy imports, and the amount spent in the fourth round is $14.89, that is, 0.53 × $28.09.

A graph is shown. The y-axis is labeled Cumulative Increase in Aggregate Expenditure from Original Increase of $100 in Government Spending. The x-axis is labeled Round. The y-axis has tick marks at 50, 100, 150, 200, and 250. The x-axis has tick marks at 5, 10, 15, 20, 25, 30, and 35. A line extends steeply upward from the origin and begins leveling off at 100, and even more at 150, then becomes almost completely level around 200 and continues approaching 250 as x approaches infinity.

Figure 11.16 The Multiplier Effect An original increase of government spending of $100 causes a rise in aggregate expenditure of $100. But that $100 is income to others in the economy, and after they save, pay taxes, and buy imports, they spend $53 of that $100 in a second round. In turn, that $53 is income to others. Thus, the original government spending of $100 is multiplied by these cycles of spending, but the impact of each successive cycle gets smaller and smaller. Given the numbers in this example, the original government spending increase of $100 raises aggregate expenditure by $213; therefore, the multiplier in this example is $213/$100 = 2.13.

The original increase in aggregate expenditure from government spending is	$100.
This is income to people throughout the economy, who pay 30 percent in taxes, save 10 percent of after-tax income, spend 10 percent of income on imports. The second-round increase is	70 – 7 – 10 = $53.
This is $53 of income to people through out the economy, who pay 30 percent in taxes, save 10 percent of after-tax income, and spend 10 percent of income on imports. The third-round increase is	37.1 – 3.71 – 5.3 = $28.09.
This is $28.09 of income to people through out the economy, who pay 30 percent in taxes, save 10 percent of after-tax income, and spend 10 percent of income on imports. The fourth-round increase is	19.663 – 1.96633 – 2.809 = $14.89.

Table 11.7 Calculating the Multiplier Effect

Thus, over the first four rounds of aggregate expenditures, the impact of the original increase in government spending of $100 creates a rise in aggregate expenditures of $100 + $53 + $28.09 + $14.89 = $195.98. Figure 11.16 shows these total aggregate expenditures after these first four rounds, and then the figure shows the total aggregate expenditures after 30 rounds. The additional boost to aggregate expenditures shrinks during each round of consumption. After about 10 rounds, the additional increments are very small indeed. After 30 rounds, the additional increments in each round are so small they have no practical consequence. After 30 rounds, the cumulative value of the initial boost in aggregate expenditure is approximately $213. Thus, the government spending increase of $100 eventually, after many cycles, produced an increase of $213 in aggregate expenditure and real GDP. In this example, the multiplier is $213/$100 = 2.13.

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.

Spending multiplier = 1/(1 – MPC × (1 – tax rate) + MPI)

The data from Figure 11.16 and Table 11.7 are

marginal propensity to save (MPS) = 30 percent,
tax rate = 10 percent, and
marginal propensity to import (MPI) = 10 percent.

The MPC is equal to 1 – MPS, or 0.7. Therefore, the spending multiplier is

\begin{array}{rcl} spending multiplier & = & \frac{1}{1 – (0.7 – (0.10)(0.7) – 0.10)} \\ = & \frac{1}{0.47} \\ = & 2.13. \end{array}

A change in spending of $100 multiplied by the spending multiplier of 2.13 is equal to a change in GDP of $213. Not coincidentally, this result is exactly what was calculated in Figure 11.16 after many rounds of expenditures cycling through the economy.

The size of the multiplier is determined by the proportion of the marginal dollar of income that goes into taxes, saving, and imports. These three factors are known as leakages, because they determine how much demand leaks out in each round of the multiplier effect. If the leakages are relatively small, then each successive round of the multiplier effect will have larger amounts of demand, and the multiplier will be high. Conversely, if the leakages are relatively large, then any initial change in demand will diminish more quickly in the second, third, and later rounds, and the multiplier will be small. Changes in the size of the leakages—a change in the marginal propensity to save, the tax rate, or the marginal propensity to import—changes the size of the multiplier.

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 2.13, then the increase in government spending to achieve that goal would be $100/2.13 = $47. Government spending of approximately $47, when combined with a multiplier of 2.13, which is, remember, based on the specific assumptions about tax, saving, and import rates, produces an overall increase in real GDP of $100, restoring the economy to potential GDP of $800, as Figure 11.17 shows.

Figure 11.17 The Multiplier Effect in an Expenditure-Output Model The power of the multiplier effect is that an increase in expenditure has a larger increase on the equilibrium output. The increase in expenditure is the vertical increase from AE₀ to AE₁. However, the increase in equilibrium output, shown on the horizontal axis, is clearly larger.

The multiplier effect is also visible on the Keynesian cross diagram. Figure 11.17 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 taxes are 30 percent of income, savings are 0.1 of after-tax income, and imports are 0.1 of before-tax income. At AE₁, the aggregate expenditure function is moved upward to reach potential GDP.

Now, compare the vertical shift upward in the aggregate expenditure function, which is $47, 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 11.15, 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 mathematical calculation.

The multiplier does not just affect government spending; it also 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 reduce aggregate expenditures and then have an even larger effect on real GDP because of the multiplier effect. Read the following Clear It Up feature to learn how the multiplier effect can be applied to analyze the economic impact of professional sports.

Clear It Up

How can the multiplier be used to analyze the economic impact of professional sports?

Attracting professional sports teams and building sports stadiums to create jobs and stimulate business growth is an economic development strategy adopted by many communities throughout the United States. In his recent article, “Public Financing of Private Sports Stadiums,” James Joyner (2012) of Outside the Beltway looked at public financing for National Football League teams. Joyner’s findings confirm the earlier work of John Siegfried of Vanderbilt University and Andrew Zimbalist of Smith College.

Siegfried and Zimbalist (2000) used the multiplier to analyze this issue. They considered the amount of taxes paid and dollars spent locally to determine whether there was a positive multiplier effect. Because most professional athletes and owners of sports teams are rich enough to owe a lot of taxes, let’s say that 40 percent of any marginal income they earn is paid in taxes. Because athletes are often high earners with short careers, let’s assume they save one third of their after-tax income.

However, many professional athletes do not live year-round in the city in which they play, so let’s say that one half of the money they do spend is spent outside the local area. One can think of spending outside a local economy, in this example, as the equivalent of imported goods for the national economy.

Now, consider the impact of money spent at local entertainment venues other than professional sports. Although the owners of these other businesses may be comfortably middle-income earners, few of them are in the economic stratosphere of professional athletes. Because their income is lower, so are their taxes. Let's say they pay only 35 percent of their marginal income in taxes. They do not have the same ability, or need, to save as much as professional athletes, so let’s assume their marginal propensity to consume is just 0.8. Finally, because more of them live locally, they will spend a higher proportion of their income on local goods—say, 65 percent.

If these general assumptions hold true, then money spent on professional sports has less local economic impact than money spent on other forms of entertainment. For professional athletes, out of a dollar earned, 40¢ goes to taxes, leaving 60¢. Of that 60¢, one third is saved, leaving 40¢, and half is spent outside the area, leaving 20¢. Only 20¢ of each dollar is cycled into the local economy in the first round. For locally owned entertainment, out of a dollar earned, 35¢ goes to taxes, leaving 65¢. Of the rest, 20 percent is saved, leaving 52¢, and of that amount, 65 percent is spent in the local area, so that 33.8¢ of each dollar of income is recycled into the local economy.

Siegfried and Zimbalist make the plausible argument that, within their household budgets, people have a fixed amount to spend on entertainment. If this assumption holds true, then money spent attending professional sports events is money not spent on other entertainment options in a given metropolitan area. Because the multiplier is lower for professional sports than for other local entertainment options, the arrival of professional sports to a city would reallocate entertainment spending in a way that causes the local economy to shrink, rather than to grow. Thus, their findings seem to confirm what Joyner reports and what newspapers across the country are reporting. A quick Internet search for “economic impact of sports” yields numerous reports questioning this economic development strategy.

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 tends to be magnified substantially so the economy becomes more unstable. With a low multiplier, in contrast, changes in aggregate demand are not multiplied much, so the economy tends to be more stable.

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

11.3 The Expenditure-Output or Keynesian Cross Model

Learning Objectives