# The Keynesian model in a closed economy

Before introducing international trade, it may be useful to review briefly the closed-economy Keynesian model, in part because it is no longer included in some introductory or intermediate macroeconomics courses.

Our economy is assumed to have two sectors: a business sector and households. We assume for the time being that it has no government, meaning no taxes or public expenditures, and no transactions with the rest of the world, and that prices remain unchanged. The model will, of course, soon be extended to include international trade. If the government were included, its expenditures would be an additional source of demand for goods and services, and taxes would be a drain on demand because they reduce consumption spending power well below earned incomes.

### Determination of the level of income

The gross national product of our economy is defined as the money value of all final products (goods and services) produced in a period of time, usually a year. This product can be divided into two categories, consumption (C) and investment (I). Thus we have the following definitional equation:

where Y stands for GNP.

In the production of goods and services making up the GDP, an equal amount of income is generated in the form of wages, rent, interest, and profit. All income earned is either spent for consumption or saved. Thus we have another definitional relation to state the disposition of income:

Y = C + S (2) Setting equations (1) and (2) equal to each other, we obtain:

18 - Open macroeconomics with fixed exchange rates 405

Subtracting C from both sides yields the important identity which states that savings equals investment:

Equations (1), (2), and (3) express ex post, or realized, relationships. They hold true, by definition, for any past period. I is actual investment, which may contain an unintended component in the form of the accumulation of unsold inventories. Intended investment equals savings only when the economy is in equilibrium.

The amount of investment expenditure is assumed to be exogenously determined (i.e. it is independent of the level of income).

Consumption, on the other hand, is a function of income: when income increases, consumption also increases, but not by as much as the increase in income. This gives us a relationship (a "consumption function") such as the following:

where Ca is the amount of consumption expenditure that is not a function of income, and c is the fraction of extra income (0 < c < 1) that is spent on additional consumption. This fraction (c) is the marginal propensity to consume, defined as

the change in C divided by the change in Y. For convenience we will assume that the marginal propensity to consume is a constant fraction.

We can obtain an expression for the equilibrium level of income by substituting equation (4) into (1), as follows:

Equation (6) states that the equilibrium level of income is equal to a multiplier [1/(1 - c)] times autonomous consumption plus investment.

A numerical example can be used to illustrate the determination of the equilibrium level of income. We assume the following consumption function:

where Ca = 50, and c = 0.60. Thus we assume that 60 percent of any increase in income will be spent for consumption.

This relationship is depicted in Figure 18.1a, which also shows the determination of Y for a given amount of investment. The consumption function, C = 50 + 0.60Y, shows how much is spent for consumption (vertical axis) at various levels of income (horizontal axis). The slope of the consumption function represents the marginal propensity to consume, c = AC/AY= 0.60. The 45° line in Figure 18.1a is a geometric device, which represents all points which are equidistant from the vertical and horizontal axes; thus the level of income can be measured either horizontally or vertically. Since all income is either spent for consumption or saved, the vertical difference between the consumption function (labeled C) and the 45° line represents the amount of saving at any level of income. At point B, where the two lines intersect, all income is spent for consumption; hence saving equals zero. At lower levels of income, saving is negative - that is, people are dis-saving, or dipping into past savings in order to spend more than their current incomes.

Figure 18.1 Equilibrium in a closed economy: (a) Y = C + I, (b) S = I. The top half of this graph, which presents the standard "Keynesian cross" diagram, indicates that equilibrium output is 200 because only at that level does total demand for goods and services, measured vertically, equal total output, which is measured horizontally. The bottom half of the figure illustrates that, at this equilibrium level of income, savings equals intended investment.

Figure 18.1 Equilibrium in a closed economy: (a) Y = C + I, (b) S = I. The top half of this graph, which presents the standard "Keynesian cross" diagram, indicates that equilibrium output is 200 because only at that level does total demand for goods and services, measured vertically, equal total output, which is measured horizontally. The bottom half of the figure illustrates that, at this equilibrium level of income, savings equals intended investment.

Given the amount of planned investment expenditures, which is assumed to be the same for all levels of income, we can now draw a line representing total expenditures (C + I) for every level of income. In Figure 18.1a, we assume I = 30, and that amount is added vertically to the consumption function to give us the C + I line, also called the "aggregate expenditure function." The equilibrium level of income is that level at which aggregate expenditure just equals the level of income as indicated by the 45° line. In Figure 18.1a, the C + I line intersects the 45° line at E, indicating an equilibrium level of income of 200. It is clear that only one such point exists: at lower levels of Y, aggregate expenditure (C + I) is above the 45° guideline; at higher levels of Y, aggregate expenditure is below the 45° guideline.

The solution can also be obtained by substituting equation (7) into equation (1), setting I = 30, and solving, as follows:

The equilibrium level of income may also be defined as the level at which intended investment just equals the amount of saving people are willing to take out of income. In Figure 18.1b, we show the saving function (S), obtained from the upper part of the diagram by taking the vertical difference between consumption at the 45° line at each level of income. The saving function can also be obtained by substituting equation (7) into equation (2), as follows:

The saving function shows that saving increases as income increases. Equation (8) indicates that 40 percent of any increase in income will be saved. The fraction, 0.40, is the marginal propensity to save, defined as s=

AS AY

As noted earlier, we assume that there are no taxes so that all income is either spent for consumption or saved. Thus it is clear that the marginal propensities to consume and save add up to 1.00, that is:

In our example, of each \$1.00 of additional income, \$0.60 will be spent for consumption and \$0.40 will be saved.

The level of planned investment is shown in Figure 18.1b by a horizontal line at I = 30. The equilibrium level of income, at which S = I, is indicated by point E, where Y = 200.

Algebraically, this solution entails substituting equation (8) into equation (3) and setting I = 30, as follows:

0.40

The two parts of Figure 18.1 contain the same information and thus yield the same outcome, although the lower part is especially useful for the case of an open economy, as we will see.

### The multiplier in a closed economy

We are now in a position to explain how a change in investment expenditure (actually, any autonomous change in expenditure) will affect the level of income, consumption, and saving. To continue the given example, suppose planned investment increases by 10. This change appears as an upward shift in the aggregate demand function) to (C + I') in Figure 18.2a, and as an upward shift in the horizontal investment line (to I') in Figure 18.2b. In both diagrams we see that the equilibrium level of income rises by 25, from 200 to 225. Thus income rises by a multiple of 2/2 times the initial increase in investment (25 10 = 2/2).

The size of this multiplier is determined by the division of an increment to income between consumption and saving - that is, the value of the marginal propensities to consume and save. In this case, with c = 0.60, when investment rises by 10, thus generating an initial increase in income of 10, 60 percent of that increase in income is spent for consumption. Therefore the first-round increase in consumption is 6. That increase in consumer expenditure is income to those who produce and sell consumer goods, and they in turn spend 60 percent of their increased income, so in the second round AC = 6 X (60%) = 3.6. This process generates a sequence:

AY = 10 + 10(0.60) + 10(0.60)2 + . AY = 10(1 + 0.60 + 0.602 + ...)

90 80

40 30

40 30

 1 1 | | ! E s i' 100 1 - —' te i -r i i ^^^^^^ 200 225

Figure 18.2 The multiplier in a closed economy. Continuing from the previous figure, if intended investment increases, C + I shifts up to C + I' in the top half of the figure and I shifts up to I' in the bottom half, both producing an increase in output which is based on the multiplier process. This is based on the marginal propensity to consume, which is the slope of the C line and therefore the C + I line.

More generally:

1 - c where c is the marginal propensity to consume. The multiplier is the expression in parentheses:

Since c + s = 1, we can replace (1 - c) in the denominator and write the multiplier as