CHAPTER 11: MONEY, INTEREST, AND INCOME
Conceptual Problems
1. The Keynesian model of income determination in Chapter 10 assumed that both prices and interest rates were fixed. But the IS-LM model in this chapter lets the interest rate fluctuate and determines the combination of output demanded and the interest rate for a fixed price level. The upward-sloping AD-curve (the [C+I+G+NX]-line in the Keynesian cross diagram) in Chapter 10 assumes fixed interest rates and prices, while the downward-sloping AD-curve that is derived at the end of Chapter 11 from the IS-LM model lets interest rates and prices fluctuate. The AD-curve derived in Figure 11-13 describes all combinations of the average price level and the level of output demanded at which the goods and money sector are simultaneously in equilibrium.
2.a. If the expenditure multiplier (a) becomes larger, the increase in equilibrium income caused by a unit change in intended spending also becomes larger. Assume investment spending increases due to a change in the interest rate. If the multiplier a becomes larger, any increase in spending will cause a larger increase in equilibrium income. This means that the IS-curve will become flatter as the size of the expenditure multiplier becomes larger.
If aggregate demand becomes more sensitive to interest rates, any change in the interest rate will cause the [C+I+G+NX]-line to shift up by a larger amount and, given a certain size of the expenditure multiplier a, this will increase the level of equilibrium income by a larger amount. As a result, the IS-curve will become flatter.
2.b. Monetary policy changes affect interest rates. This leads to a change in intended spending, which is reflected in a change in income. In 2.a. it was noted that a steep IS-curve means either that the expenditure multiplier a is small or that intended spending is not very interest sensitive. Assume now that the central bank undertakes expansionary monetary policy, which will reduce interest rates. If intended spending is not very interest sensitive, this will result in only a small increase in aggregate demand. Similarly, if the expenditure multiplier is small, a change in spending will not significantly affect output. With a steeper IS-curve, monetary policy changes have a weaker effect on equilibrium output.
3. Assume for simplicity that money supply always remains fixed. Any increase in income will increase money demand and the resulting excess demand for money will drive the interest rate up. This, in turn, will reduce the quantity of money balances demanded to the point where the money sector is back in equilibrium. But if money demand is not very interest sensitive, a larger increase in the interest rate is needed to reach a new equilibrium in the money sector. As a result, the LM-curve becomes steeper.
Along the LM-curve, an increase in the interest rate is always associated with an increase in income. This means that an increase in money demand (a shift of the money demand curve to the right due to an increase in income) has to be offset by a decrease in the quantity of money demanded (a movement along the money demand curve from right to left caused by an increase in the interest rate) to keep the money sector in equilibrium. But if money demand becomes more income sensitive, the shift in the money demand curve associated with a specific change in income will be larger and a larger increase in the interest rate is required to bring the money sector back into equilibrium. Therefore the LM-curve becomes steeper as money demand becomes more income sensitive.
4.a. A horizontal LM-curve implies that the public is willing to hold whatever money is supplied at any given interest rate. In such a case, changes in income will not affect the equilibrium interest rate in the money sector. But if the interest rate remains constant, we are back to the analysis of the simple Keynesian model used in Chapter 10. In other words, there is no offsetting effect (or crowding-out effect) to fiscal policy.
4.b. A horizontal LM-curve implies that changes in income do not affect interest rates in the money sector. Therefore, if expansionary fiscal policy is implemented, the IS-curve shifts to the right, but the interest rate remains constant. In this case, the level of investment spending is not negatively affected and there is no crowding-out effect. In terms of Figure 11-3, the interest rate no longer serves as the link between the goods and asset markets.
4.c. A horizontal LM-curve results if the public is willing to hold whatever money balances are supplied at a given interest rate. This situation is called the liquidity trap. Similarly, if the Fed is prepared to peg the interest rate at a certain level, then any change in income will always be accompanied by an appropriate change in money supply. This will lead to continuous shifts in the LM-curve, which is equivalent to having a horizontal LM-curve, since the interest rate is never allowed to change.
5. From the material presented in the text we know that when intended spending becomes more interest sensitive, the IS-curve becomes flatter. Now assume that an increase in the interest rate stimulates saving and thus reduces consumption. But even if saving is not affected by a change in the interest rates, most likely consumption on durable goods will be reduced if interest rates rise. This means that now not only investment spending but also consumption is negatively affected by an increase in the interest rate. In other words, the [C+I+G+NX]-line in the Keynesian cross diagram now shifts down further than previously and the level of equilibrium income decreases more than before. In other words, the IS-curve becomes flatter.
This can also be shown algebraically, since we can now write the consumption function in the following way:
C = Co + cYD - gi
In a simple model of the expenditure sector without income taxes, the equation for aggregate demand will now be
AD = Ao + cY - (b + g)i.
From Y = AD ==> Y = [1/(1 - c)][Ao - (b + g)i] ==> i = [1/(b + g)]Ao - [(1 - c)/(b + g)]Y
Therefore, the IS-curve now becomes flatter as its slope has been reduced from (1 - c)/b to (1 - c)/(b + g).
6. In the IS-LM model, a simultaneous decline in the interest rate and the level of output can only be caused by a shift of the IS-curve to the left. This shift could easily have been caused by a decrease in private spending due to negative business expectations or a decline in consumer confidence. In 1991, the economy was in a recession and firms did not want to invest in new machinery since they did not want to be left holding unwanted inventory. Since consumer confidence was very low and people feared lay-offs, consumer spending decreased also. In the IS-LM diagram below, the adjustment process can be described as follows:
Io ¯ ==> Y ¯ (the IS-curve shifts left) ==> md ¯ ==> i ¯ ==> I ==> Y .
Effect: Y ¯ and i ¯ .
i ISo LM
IS1
i1
i2
0 Y2 Y1 Y
Technical Problems
1.a. Each point on the IS-curve represents an equilibrium in the expenditure sector. (Note that this is a closed economy, that is, NX = 0). The IS-curve can be derived by setting actual income equal to intended spending, or
Y = C + I + G = (0.8)[1 - (0.25)]Y + 900 - 50i + 800 = 1,700 + (0.6)Y - 50i ==>
(0.4)Y = 1,700 - 50i ==> Y = (2.5)(1,700 - 50i)
==> Y = 4,250 - 125i. IS-curve
1.b. The IS-curve shows all combinations of the interest rate and the output level such that the expenditure sector (the goods market) is in equilibrium, that is, actual output equals intended spending. A decrease in the interest rate stimulates investment spending, making intended spending greater than actual output. The resulting unintended inventory decrease leads firms to increase their production until actual output is again equal to intended spending. This means that the IS-curve is downward sloping.
1.c. Each point on the LM-curve represents an equilibrium in the money sector. Therefore the LM-curve can be derived by setting real money supply equal to real money demand, that is,
M/P = L ==> 500 = (0.25)Y - 62.5i ==> Y = 4(500 + 62.5i)
==> Y = 2,000 + 250i. LM-curve
1.d. The LM-curve shows all combinations of the interest rate and level of output such that the money sector is in equilibrium, that is, the demand for real money balances is equal to the supply of real money balances. An increase in income will increase the demand for real money balances. Given a fixed real money supply, this will lead to an increase in interest rates, which will then reduce the quantity of real money balances demanded until the money sector is again in equilibrium. In other words, the LM-curve is upward sloping.
1.e. The equilibrium levels of income and the interest rate are determined by the intersection of the IS-curve with the LM-curve. At this point, the expenditure sector and the money sector are both in equilibrium simultaneously.
From IS = LM ==> 4,250 - 125i = 2,000 + 250i ==> 2,250 = 375i ==> i = 6
==> Y = 4,250 - 125*6 = 4,250 - 750 ==> Y = 3,500
Check: Y = 2,000 + 250*6 = 2,000 + 1,500 = 3,500
i
125 IS
LM
6
0
2,000 3,500 4,250 Y
2.a. As we have seen in 1.a., the value of the expenditure multiplier is a = 2.5. This multiplier is derived in the same way as in Chapter 10. But now intended spending also depends on the interest rate, so we no longer have Y = aAo, but rather
Y = a(Ao - bi) = (1/[1 - c + ct])(Ao - bi) ==> Y = (2.5)(1,700 - 50i) = 4,250 - 125i.
2.b. In the IS-LM model, an increase in government purchases (G) will have a smaller effect on output than in the model of the expenditure sector used in Chapter 10, in which interest rates are assumed to be fixed. This can be demonstrated most easily with a numerical example. If government purchases are increased by DG = 300, the IS-curve shifts parallel to the right by
DIS = (2.5)(300) = 750.
Therefore, the equation of the new IS-curve is: Y = 5,000 - 125i.
From IS' = LM ==> 5,000 - 125i = 2,000 + 250i ==> 375i = 3,000 ==> i = 8
==> Y = 2,000 + 250*8 ==> Y = 4,000 ==> DY = 500
When interest rates are assumed to be fixed, the size of the expenditure multiplier is a = 2.5, that is, (DY)/(DG) = 750/300 = 2.5. However, when interest rates are allowed to vary, the size of the multiplier is reduced to a1 = (DY)/(DG) = 500/300 = 5/3 = 1.67.
2.c. An increase in government purchases by DG = 300 causes a change in the interest rate from io = 6 to i1 = 8, that is, by 2 percentage points. Therefore government spending has to change by DG = 150 to increase the interest rate by one percentage point.
2.d. The simple multiplier a = 2.5 in 2.a. shows the magnitude of the horizontal shift in the IS-curve, given a change in autonomous spending by one unit. But an increase in income increases money demand and this leads to an increase in the interest rate. The higher interest rate crowds out some investment spending and this has a dampening effect on the level of output. The multiplier effect in 2.b. is therefore smaller than the multiplier effect in 2.a., and has been reduced to a1 = 1.67.
3.a. An increase in the income tax rate (t) will reduce the size of the expenditure multiplier (a). But as the expenditure multiplier becomes smaller, the IS-curve becomes steeper. As we can see from the equation for the IS-curve, a change in the income tax rate (t) will cause a rotation around its vertical intercept, as is shown below:
Y = a(Ao - bi) = [1/(1 - c + ct)](Ao - bi)
==> i = (1/b)Ao - (a/b)Y = (1/b)Ao – [(1 - c + ct)/b]Y.
In other words, as the income tax rate (t) gets larger, the slope of the IS-curve gets steeper, while the vertical intercept remains unaffected.
3.b. If the IS-curve shifts to the left and becomes steeper, the equilibrium income level will decrease. A higher tax rate will reduce private spending and this will lead to a lower level of national income.
3.c. i IS1
IS2 LM
i1
i2
0
Y2 Y1 Y
As we can see from the diagram below, the equilibrium level of income and the interest rate both decrease when the income tax rate is increased. The adjustment to the new equilibrium can be expressed as follows:
t ↑ ==> C ↓ ==> Y ↓ ==> md ↓ ==> i ↓ ==> I ↑ ==> Y ↑. Effect: Y ¯ and i ¯
4.a. If money demand is less interest sensitive, then the LM-curve is steeper and monetary policy changes have a stronger effect on equilibrium income. If money supply is changed to a new fixed level, the adjustment to a new equilibrium in the money sector has to come solely through changes in money demand. If money demand is less interest sensitive, any increase in money supply requires a larger increase in income and a larger decrease in the interest rate in order to bring the money sector into a new equilibrium.
i i
IS LM1 LM2 IS
LM1
i1 i1 LM2
i2
i2
0 Y1 Y2 Y 0 Y1 Y2 Y
The adjustment process in each of the two diagrams is the same, but in the case of a more interest-sensitive money demand (a flatter LM-curve), the change in Y and i will be smaller.
176
(M/P) ↑ ==> i ↓ ==> I ↑ ==> Y ↑ ==> md ↑ ==> i ↑ Effect: Y and i ¯
Section 11-5 derives the equation for the LM-curve as
i = (1/h)[kY - (M/P)]
and the equation for the monetary policy multiplier as
(DY)/D(M/P) = (b/h)g.
If money demand becomes more interest sensitive, the value of h becomes larger and the slope of the LM-curve becomes flatter, while the size of the monetary policy multiplier becomes smaller.