A simple income–expenditure model
Because accounting identities—between gross national product and gross national income, between saving and investment, and so on—express relationships that must hold whatever the level of income, they cannot be used to explain what determines the particular level of income in a given period or what causes the level of income to change from one period to the next. The explanation of what happens must be based on statements about the behaviour of the participants in the economic system; in the present context, this means the behaviour of firms and households.
The following oversimplified model of an economy assumes that the business sector will be satisfied to maintain any given level of output as long as aggregate demand (that is, expenditures on final goods) exactly equals the volume of income generated at that level of output. If, in a given period, aggregate demand exceeds the income payments made by firms in producing that period’s output, firms will be expanding in the next period; if aggregate demand falls short of the income payments made, firms will contract in the next period. The naïveté of this supply hypothesis is evident from the fact that the behaviour of firms is described without any reference to the costs of their inputs or to the price of their outputs; the business sector passively adapts output and income generated to the level of aggregate demand. In this model, the level of income is entirely determined by aggregate demand. Firms will act so as to maintain that income flow if, and only if, the exact same amount that they pay out as incomes “comes back to them” in the form of spending on final goods output. If aggregate demand shrinks, production and employment will decline and there will be downward pressure on the price level; if aggregate demand swells, there will be an inflationary problem.
In the system of Figure 1, all of the income generated accrues to households. Households allocate their income to consumption and saving. With consumption there is no problem—it constitutes spending on final goods. Saving, however, does not constitute spending on final goods output. This part of the income generated by the business sector does not automatically come back to it in the form of revenue from sales. Saving, therefore, may be treated as a leakage from the circular flow.
Investment, which consists of spending of capital by the business sector on new plant and equipment and on desired additions to inventories, is, in the same terminology, an injection into the circular flow. If, for example, investment and saving each amount to $20 million per year, the leakage and the injection will balance. But if saving is $20 million per year and the injection of investment expenditures is only $10 million per year, there will be a disequilibrium. Unsold goods will accumulate at an annual rate of $10 million. The business sector, however, will not rest content with this state of affairs but will act to reduce output, employment, and (perhaps) prices. Households will be forced to reduce their consumption spending. The reduction of income will go on until the planned (or desired) rates of saving and investment become equal. A similar argument will show that, if the leakage of planned saving were to fall short of the injection of planned investment, the level of income would rise.
When income is at a level such that there is no ongoing tendency for it to change in either direction, the system is in “income equilibrium.” The simple system depicted in Figure 1 is in income equilibrium when the condition shown by this equation is fulfilled: I = S. This is not, however, the accounting identity discussed earlier. The symbols I and S now refer to planned, or desired, magnitudes, which may very well be unequal. When planned investment exceeds planned saving, income will be rising. When planned saving exceeds planned investment, income will be falling. An equivalent way of stating the above “equilibrium condition” is to write Y = C + I. In this equation the left-hand side is actual income and the right-hand side is planned aggregate demand.
This is the simplest class of income-determination model. It makes no allowance for international trade or government economic activity. Those may be treated in the same way that saving and investment were treated—as leakages or injections. Thus, exports constitute spending by foreign nationals on domestic goods—an injection. Imports constitute spending out of domestic income on foreign goods—a leakage. Taxes are taken out of the circular flow—a leakage—whereas government expenditures are an injection. The effects of these leakages and injections on the level of income are analogous to those of saving and investment. If income is initially at an equilibrium level, an increase in a leakage (if not at the same time offset by a decrease in another leakage or an increase in an injection) will cause income to fall. An increase in an injection (not offset by a decrease in another injection or an increase in a leakage) will cause income to rise. An income equilibrium is reached when the sum of all leakages is balanced by the sum of all injections.
The multiplier
The simple income–expenditure model of the economy is not a complete model. It suffices to show only the direction of the change in income that would result from, say, a decline in planned investment (or a rise in taxes or a decline of exports). It does not show the extent of the income change.
To do this the model must be expanded to include a description of how consumers spend their incomes. For the sake of the exposition, one may assume that the spending of households varies according to the size of their incomes. A simple way of putting this is the following equation: C = a + by. In this equation the coefficient a is a constant indicating the amount that households will spend on consumption independently of the level of income received in the current period, and the coefficient b gives the fraction of each dollar of income that will be spent on consumption goods.
If one were able to obtain reliable quantitative information on the volume of investment spending being planned and on the coefficients a and b of the “consumption function” above, one could then calculate the value of aggregate demand (C + I) for every possible level of income Y. Only one of these alternative levels of income is an equilibrium one; that is, one for which aggregate demand will ensure that all of the income paid out by firms “comes back” to the business sector as spending on final goods. The equilibrium condition is: Y = C + I.
Figure 2 shows how the level of income in the system is determined, on the assumption that investment is $20 million, that the coefficient a is $20 million, and that the coefficient b (the fraction of each dollar of income that consumers will spend) is 0.6. The horizontal axis measures income, the vertical, aggregate demand (C + I). The line drawn at a 45° angle (from 0) contains all of the points at which suppliers might be in equilibrium; i.e., the points in the space at which aggregate demand would have the same value as income. The investment schedule (marked I = Ī0) is drawn parallel to the income axis at height 20, showing that investment spending does not depend on income. The consumption function (marked C = a + by) starts at 20 on the vertical axis (the value of a) and rises 60 cents for each dollar of income (the value of b) to the right. The aggregate demand schedule (marked C + Ī0) is obtained by the vertical summation of the C and Ī0 schedules. It contains all of the points at which demanders would be in equilibrium, showing, for each level of income, the volume of spending on final goods that they would be satisfied to maintain.
The only position that demanders and suppliers will both be satisfied to maintain is given by the intersection of the aggregate demand schedule with the 45° line. In Figure 2 this point (Ŷ0) is found at an income level of $100 million. For this simple system, which has but one leakage and one injection, the equilibrium level of income may equally well be regarded as determined by the condition that planned saving equals planned investment. Since saving is defined as household income not spent on consumption (i.e., Y - C ≡ S), one obtains (by substituting a + by for c) the saving schedules S = -a + (1 - b) Y, which in Figure 2 is shown to intersect the investment schedule at Y = $100 million.
Figure 2 shows what will happen if this equilibrium is disturbed. Consider a (temporary) situation in which income is running at more than $100 million per year. At all levels of income to the right of Ŷ0 aggregate demand (C + Ī0) is seen to fall below supply as given by the 45° line. (Also, saving exceeds investment.) The business sector will not be willing to maintain this state of affairs but will contract. An excess supply of final goods is associated with falling income. Similarly, at income levels to the left of Ŷ0, where investment exceeds saving, aggregate demand will exceed supply. An excess demand for final goods is associated with rising income.
Finally, Figure 2 shows how much income would fall as a result of a decline in investment by $10 million per year (cf. the dotted lines). The decline in investment is shown by the shift of the investment schedule from Ī0 to Ī1, which results in a downward shift of the aggregate demand schedule from C + Ī0 to C + Ī1. The new income equilibrium (Ŷ1) is found at Y = $75 million.
Thus, a change in investment spending (ΔI) of $10 million is found to lead to a change in income (ΔY) of a larger amount, here $25 million, which is to say, by a multiple of 2.5. The reason is that, when the $10 million is transmitted to households as income, households will increase their consumption spending by $6 million (b × $10 million). This rise in consumption spending again raises income, and of this additional income 60 percent is also spent on consumption—and so on. Each time, 40 percent of the increment to income “leaks” into saving. The relationship between the initial change in “autonomous spending” (ΔI) and the change in the level of income (ΔY), which will have taken place once this process has run its course, is given by: 
where, following Keynes, the expression (1/1 - b) is called the “Multiplier.”
The model of income determination presented above is exceedingly simple; it captures little of the complexity of a modern industrialized economy. It does, however, suggest one approach to the problem of stabilizing the economy at a high level of income and employment. Assuming that the consumption function is fairly stable (i.e., that the level of consumption spending associated with any level of income can, with a fair degree of accuracy, be predicted on the basis of past experience), fluctuations in income may be attributed to changes in the other variables. Historical statistics show investment spending by private business to have been the most volatile of the major components of national income; changes in investment, therefore, tend (as in the example above) to be the focus of concern for one school of economists. The implication is that the government can manipulate “injections” and “leakages” so as to offset changes in private investment. Thus, a drop in investment might be offset by a corresponding increase in government expenditures (increasing an injection) or a decrease in taxes (decreasing a leakage). These measures belong to fiscal policy.