E = E{Y, G, T)

E=E(Y, i-7T«, G, T)

FIGURE 5.3 The Keynesian cross

actual expenditure equals the economys output, Y. In equilibrium, planned and actual expenditures must be equal. If planned expenditure falls short of actual expenditure, for example, firms are accumulating unwanted inventories; they will respond by cutting their production. Thus equilibrium requires

E = Y.

Substituting (5.6) into (5.4) yields

Y = E(Y,i -tt.G.T).

(5.6)

(5.7)

Figure 5.3, the Keynesian cross, depicts equations (5.4) and (5.6) in (Y,E) space for a given level of the interest rate. Equation (5.6) isjust the 45-degree line. Since planned expenditure increases less than one-for-one with , the set of points satisfying (5.4) is less steep than the 45-degree line. The point where the planned expenditure curve crosses the 45-degree line (Point A) shows the unique level of income where actual and planned expenditures are equal for the given interest rate.

The Keynesian cross is sometimes described as a theory of income determination. But this is correct only if the interest rate can be treated as ftxed, which is often inappropriate. Thus it is better to think of the Keynesian cross as an ingredient of a larger model.

(5.8)

li 1 - £y

Since this is an expression for dY/di (rather than di /dY), it implies that the /5 curve is flatter when cither E, or £y is larger. Intuitively, the larger the effect of the interest rate on planned expenditure, the larger the downward shift of the planned expenditure line, and thus the larger the fall in output. Similarly, the steeper the planned expenditure line, the more output must fall in response to a given downward shift of the planned expenditure line to reach a point where planned and actual expenditures are again in balance, and thus the larger the faU in output. This last effect is the famous multi-p/zer: because £ depends on Y, the fall in Y needed to restore the equality of E and Y is larger than the amount that E falls at a given Y.

The AD Curve

The intersection of the /5 and LM curves shows the values of z and Y such that the money market clears and actual and planned expenditures are equal for given levels of P, *, G, and T. To see how the 75 and IM curves imply the existence of a downward-sloping relationship between P and Y, consider the effects of assuming a higher value of P. Since the price level does not enter the planned expenditure function, £(•), the /5 curve is unaffected. The rise in the price level reduces the supply of real money balances, however. Thus a higher interest rate is needed to clear the money market for a given level of income, and so the IM curve shifts up. As a result, z rises and Y falls. This IS shown in Figure 5.4. Thus the level of output at the intersection of the IS and IM curves is a decreasing function of the price level. This is what is shown by the aggregate demand curve.

To find the slope of the AD curve, differentiate (5.1) and (5.7) with respect to P. This yields two equations in two unknowns:

M di

P2 dP dY

AD dP

AOdP

(5.9)

These can be solved to obtain

AD-dP

(5.10)

An increase in the interest rate shifts the planned expenditure line down (since £(•) is decreasing in i - ), and thus reduces the level of income at which actual and planned expenditures are equal; in terms of the diagram, an increase in the interest rate from i to i shifts the intersection of the two lines from Point A to Point A. Thus the /5 curve slopes down.

Differentiating (5.7), one can show

FIGURE 5.4 The effects of an increase in the price level

-M/P

AD [(1-£y)L,+ Ly

(5.11)

This expression is unambiguously negative, and it shows the determinants of the slope of the aggregate demand curve.

Example: The Effects of an Increase in Government Purchases

The 75 and LM curves provide a simple model of aggregate demand that can be used to analyze many issues. Suppose, for example, that government purchases rise. The increase in G raises planned expenditure for a given level of output and the interest rate. The planned expenditure line in Figure 5.3 therefore shifts up, and so the level of Y such that actual and planned expenditures are equal is higher for a given level of the interest rate. Thus the IS curve shifts to the right; this is shown in Panel (a) of Figure 5.5. The shift in the 75 curve raises Y (and i) for a given price level, and thus moves the AD curve outward; this is shown in Panel (b) of the figure.

The IS-LM diagram is drawn for a given value of P. Thus the amount that output increases in the IS-LM diagram is the same as the amount that the aggregate demand curve shifts to the right at the value of P assumed in the IS-LM diagram