## 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. 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:

 AE = C + I + G + X – M AE = 140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y

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:

Y = 140 + 0.9(Y – T) + 400 + 800 + 600 – 0.15Y

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.

 Y = 140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.15Y Y = 140 + 0.9Y –0.27Y + 1800 – 0.15Y Y = 1940 + 0.48Y Y – 0.48Y = 1940 0.52Y = 1940 $\displaystyle\frac{0.52\text{Y}}{0.52}$ = $\displaystyle\frac{1940}{0.52}$ Y = 3730

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.

 Y = 140 + 0.9(Y – 0.3Y) + 400 + 800 + 600 – 0.1Y Y = 1940 – 0.53Y 0.47Y = 1940 Y = 4127

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

 Y = 140 + 0.9(Y – 0.3Y) + 500 + 800 + 600 – 0.15Y Y = 2040 + 0.48Y Y – 0.48Y = 2040 0.52Y = 2040 Y = 3923

For issues of policy, the key questions would be 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).

 Y = 200 + 0.9(Y – 0.3Y) + 600 + 1000 + 600 – 0.1(Y – 0.3Y) Y – 0.63Y + 0.07Y = 2400 0.44Y = 2400 Y = 5454

Step 9. Assume that the full employment level of output is 6,000. What level of government spending would be 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:

6000 = 200 + 0.9(6000 – 0.3(6000)) + 600 + G + 600 – 0.1(6000 – 0.3(6000))

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 will rise from 5,454 to 6,000 can be answered without working through the algebra, just by using the multiplier formula. The multiplier equation in this case is:

$\displaystyle\frac{1}{1-0.56}=2.27$

Thus, to raise output by 546 would require 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 (like 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 will shape the model to fit the specific question 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 B.7 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 B.3 are lifted from the earlier Table B.2, 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 30% of real GDP is collected in taxes. The third column is based on an 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, sums up C + I + G + X – M. This aggregate expenditure line is illustrated in Figure B.7. Figure B.7. A Keynesian Cross Diagram Each combination of national income and aggregate expenditure (after-tax consumption, government spending, investment, exports, and imports) is graphed. The equilibrium occurs where aggregate expenditure is equal to national income; this occurs where the aggregate expenditure schedule crosses the 45-degree line, at a real GDP of $6,000. Potential GDP in this example is$7,000, so the equilibrium is occurring at a level of output or real GDP below the potential GDP level.

Table B.3. National Income-Aggregate Expenditure Equilibrium
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

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 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 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 will 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 to spending on domestic 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 can also be read off the table under the figure; it is the level of national income where aggregate expenditure is equal to national income.