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CHAPTER 11 MONEY, INTEREST, AND INCOME Chapter Outline The Goods Market and the IS Curve Investment and the Interest Rate The Slope of the IS Curve The Role of the Multiplier The Position of the IS Curve A Summary of the IS Curve The Money Market and the LM Curve The Demand for Money The Supply of Money, Money Market Equilibrium and the LM Curve The Slope of the LM Curve Shifts in the LM Curve A Summary of the LM Curve Equilibrium in the Goods and Money Markets Changes in the Equilibrium Income and Interest Rate: A First Look at Policy Deriving the Aggregate Demand Curve Working With Data Changes from the Previous Edition One of the major problems with this chapter in the previous editions is that the material was presented in a manner that made it sound like it would be difficult, which is not true. Therefore, the long introduction has been shortened, the confusing diagram (former Figure 12-2) has been removed, and the extra part concerning outline of the chapter has been removed. The derivation of the IS curve section has been rewritten at the beginning. Box 11-1 and Box 11-2 and 11-3 are new The LM curve section has been rewritten to make it more clear, and the relevant diagrams are now side by side, which makes much more sense. The last section of the chapter is rewritten to show the comparative statics of shifts in IS and LM; this serves as a good introduction to Chapter 12. Learning Objectives Students should be aware that the IS-LM model discussed here is a simplified, short-run, static macro-model, in which prices are assumed to be fixed. Students should understand that every point on the IS-curve represents an equilibrium in the expenditure sector and every point on the LM-curve represents an equilibrium in the money sector. Students should be aware that, although there are many combinations of the level of income and the interest rate that bring either the expenditure sector or the money sector into equilibrium, there is only one combination of these two variables that will bring both sectors into equilibrium simultaneously. 118 Students should be able to identify the factors that determine the slope of the IS- and LM-curves and the factors that lead to a shift in either one. Students should be able to identify the strengths of fiscal and monetary policies given different assumptions about the slopes of the IS- and LM-curves. Students should be able to graphically derive the AD-curve. Students should understand that every point on the AD-curve represents a situation in which the goods and money sectors are simultaneously in equilibrium. Achieving the Objectives Chapter 11 deals with a static, short-run macro-model in which interest rates can vary but prices are still assumed to be fixed. This is the so-called IS-LM model, which is used to analyze the interaction of the goods and money sectors. The IS-LM model gives students an understanding of interest rates and their role as an additional determinant of aggregate demand. As a result, students will be better able to assess how the composition of aggregate demand changes as interest rates fluctuate, which allows for a more comprehensive analysis of the effects of fiscal and monetary policy on output demanded. First, the IS-curve is derived as a logical extension of the Keynesian cross diagram. In the Keynesian cross diagram, interest rates are assumed to be constant. Now they are allowed to fluctuate. When interest rates decrease, it becomes more profitable to increase the existing capital stock, leading to an increase in planned investment spending. Therefore the [C+I+G+NX]-line shifts up. As intended spending exceeds actual output, unintended inventories decrease and firms respond by increasing production. This leads to an increase in the level of output to the point where actual output is again equal to intended spending. Thus it becomes obvious that the IS-curve shows the combinations of output levels and interest rates at which the goods sector is in equilibrium. It also becomes clear why the IS-curve is downward sloping: lower interest rates lead to more investment spending and therefore a higher level of output. The slope of the IS-curve is determined by two factors: the interest sensitivity of investment spending and the size of the expenditure multiplier. As investment becomes more interest sensitive, any change in the interest rate leads to a larger shift in the [C+I+G+NX]-line, leading to a higher level of equilibrium output and a flatter IS-curve. If the value of the multiplier becomes larger, then any shift in the [C+I+G+NX]-line caused by a specific change in the interest rate will cause a larger increase in equilibrium income. Again the IS-curve becomes flatter. As we saw previously, any other factor that may increase the slope of the [C+I+G+NX]-line, such as a decrease in the income tax rate, will also increase the size of the expenditure multiplier and therefore affect the slope of the IS-curve. The IS-curve shifts parallel with any change in autonomous spending (Ao). In Chapter 9, we derived the following equation: Y = (Ao), which we now can replace with IS = (Ao), where is the expenditure multiplier and IS is the value of the horizontal shift of the IS-curve. The LM-curve, which shows the combinations of interest rates and income levels where the money sector is in equilibrium, is derived in two steps. First, it is shown that the demand for real money balances (L) is positively related to the level of income (Y), since people will hold more money to finance 119 their increased expenditures. But it is also negatively related to the interest rate (i), since the interest rate is the opportunity cost of holding money rather than bonds. The equation for real money demand can therefore be written as L = kY - hi with k > 0 and h > 0. Then real money supply, that is, nominal money supply (M) divided by the price level (P) is set equal to real money demand to achieve an equilibrium in the money sector. In other words, the equation of the LM-curve is derived in the following way. (M/P) = L = kY - hi ==> i = (1/h)[kY - (M/P)]. The LM-curve is derived graphically in Figure 12-8. An increase in income will increase the demand for real money balances (a shift from L1 to L2). Since real money supply is fixed, the interest rate will increase, reducing the quantity of real money balances demanded again until the money market clears (a movement along L2 to the left). Therefore, the LM-curve is upward sloping. The supply of money is assumed to be fixed, that is, determined by the central bank. Therefore, the slope of the LM-curve is solely determined by the interest elasticity and the income elasticity of money demand. If money demand becomes less interest sensitive, then any increase in income will have to be offset by a larger increase in the interest rate to bring the money sector back to equilibrium again. This makes the LM-curve steeper. If money demand becomes more income elastic, then the interest rate has to increase more in response to a change in income to equate money demand with money supply. This also makes the LM-curve steeper. The LM-curve shifts parallel with any change in real money supply. The size of the shift can be derived as follows: i = (1/h)[kY - (M/P)] ==> Y = (1/k)[(M/P) + (h/k)i] ==> LM = (1/k)[(M/P)], with LM being the size of the horizontal shift of the LM-curve. Even though there are many combinations of interest rates and levels of income that bring either the goods sector or the money sector into equilibrium, there is only one specific combination of these two variables that will bring both of these sectors into equilibrium simultaneously. The adjustment to such an equilibrium is based on the following two assumptions: The level of output increases (decreases) whenever there is excess demand (supply) of goods and services. The interest rate rises (falls) whenever there is excess demand (supply) of real money balances. The IS-LM model that has been derived here provides a simple but effective framework for analyzing the short-run effects of fiscal and monetary policies on the demand for output and interest rates. The policy applications of this model, however, will be dealt with in the next chapter. Chapter 10 concludes with a graphical derivation of the AD-curve. The optional Section 12-5 also formally derives the fiscal and monetary policy multipliers for the IS-LM framework. The AD-curve shows all possible combinations of the price and output level at which the goods and money sectors are simultaneously in equilibrium. In the IS-LM model, the price level is assumed to be fixed. But when prices can fluctuate, the level of real money balances (M/P) varies inversely with changes in the price level (P). As the price level decreases, real money balances increase, the LM-curve 120 shifts to the right, and a new simultaneous equilibrium in the goods and money sectors is reached at a higher output level and a lower interest rate. Therefore, the AD-curve is downward sloping. The formal (algebraic) treatment of the IS-LM model shows more clearly that the fiscal policy multiplier for the IS-LM model is smaller than the simple expenditure multiplier of the previous chapter, which applied only when interest rates were assumed to be constant. This is the result of the dampening effect of rising interest rates on intended spending that is associated with fiscal expansions. Clearly, the greater the interest elasticity of investment or the smaller the interest elasticity of money demand, the less powerful fiscal policy is in stimulating intended aggregate demand. The monetary policy multiplier indicates that monetary policy is more powerful when the interest elasticity of investment increases, the interest elasticity of money demand decreases, and the fiscal policy multiplier is larger. This indicates that fiscal policy is more effective with a steep IS-curve and a flat LM-curve, while monetary policy is more effective with a flat IS-curve and a steep LM-curve. Suggestions and Pitfalls Many instructors find the IS-LM model very helpful. It allows for a discussion of the effects of monetary and fiscal policies on national income and interest rates and shows the effectiveness of fiscal and monetary policy under different conditions in a relatively simple framework. Since most of the policy applications are presented in Chapter 12, instructors may want to assign both chapters simultaneously. In spite of its simplicity, students often have difficulties with the IS-LM framework. For them it is just another graph they have to put up with and they often are reluctant to carefully examine the adjustment processes that are needed to achieve a simultaneous equilibrium in the goods and money sectors. Mathematically weak students tend to get lost when the abstract algebraic derivations of the fiscal and monetary policy multipliers are attempted, which is why Section 11-5 is marked as optional. Much confusion can be avoided if a few well-chosen and simple numerical examples are used to calculate the equations for the IS- and LM-curves, from which the equilibrium values of output and the interest rate can then be derived. By changing the value of either government purchases or money supply and calculating the new equilibrium values of output and interest rates, instructors can show the changes that occur in all the other variables included in the model. Some numerical examples are given below in "Additional Problems." Another way to present the material to those students who have a high level of math anxiety is to rely solely on the graphical derivation of the IS-LM model. It is important to take great care in separately deriving first the IS-curve and then the LM-curve graphically, each time also incorporating an example of a shift in these curves. In presenting the graphical analysis of the IS-LM model, it is important to emphasize the role of the interest rate as a link between the goods sector and the money sector. It should be stressed that while the expenditure sector and the money sector are two distinct sectors of our economy, they are nonetheless interdependent. A disturbance in either one of these sectors will affect the other in a way that makes it possible to eventually reach another simultaneous equilibrium. It is therefore important to emphasize the difference between a shift in any of the curves (caused by a change in policy or an external disturbance) and movement along any of the curves (the resulting adjustment process). Policy changes that lead to a parallel shift versus a change in the slope can also be discussed, but this should be done when the material in Chapter 13 is covered. When graphically deriving the IS-curve, a heuristic explanation should be given of the inverse relationship between the level of investment spending and the interest rate. At this point, some reference should also be made to Chapter 15, which discusses investment spending in more detail. The interest elasticity of investment greatly determines the slope of the IS-curve and thus the effectiveness of fiscal and monetary policy, in changing the level of output demanded. Therefore, instructors should carefully 121 explain not only why the IS-curve is downward sloping, but also how a change in the interest elasticity of investment affects its slope. But the slope of the IS-curve is also determined by the size of the expenditure multiplier, which, in turn, depends on the value of the marginal propensity to consume and the income tax rate. This can be shown via a graphical example, derived algebraically, or demonstrated with a simple mathematical example. A similar approach should be chosen when the LM-curve is derived. It is helpful to at least briefly explain the relationship between the amount of real money balances demanded and the level of income and the interest rate, with reference to Chapter 17, in which the demand for money is discussed in more detail. Assumptions about the interest elasticity of money demand are crucial in determining the slope of the LM-curve and therefore the effectiveness of fiscal and monetary policy. This can again be made clear with a graphical or numerical example or it can be algebraically derived. One easy way to show that the interest elasticity of money demand matters is by stating the equilibrium condition in the money sector, that is, real money demand (md) has to equal real money supply (ms). From the equation md(i,Y) = ms = M/P we can see not only that the LM-curve is upward sloping, but also that the interest elasticity of money demand will determine its slope. Any increase in income will increase the demand for real money balances. If we assume that money supply is fixed, then an equilibrium in the money sector can only be restored if the money balances demanded are reduced again through an increase in interest rates. But the more interest inelastic the demand for real money balances is, the higher the increase in the interest rate needed to bring the quantity of real money balances demanded back to the original level and the steeper the LM-curve will be. The slope of the LM-curve also depends on the income elasticity of money demand and to some extent, also the interest elasticity of money supply. As Chapter 18 will show, money supply can be influenced by the behaviour of banks or the public, and is therefore somewhat interest elastic. Given the time constraints of a normal semester, instructors may, however, want to concentrate on the interest elasticity of money demand as a factor in determining the steepness of the LM-curve. Every point on the IS-curve signifies a combination of the level of output demanded and the interest rate that brings the goods sector into equilibrium; every point on the LM-curve signifies a combination of the level of output demanded and the interest rate that brings the money sector into equilibrium. Therefore is should be quite clear that the intersection of these two curves signifies that both sectors are simultaneously in an equilibrium. In a simple IS-LM diagram, we assume that the price level is fixed. But if we now let the price level fluctuate, then real money balances fluctuate and the LM-curve shifts, leading to a new intersection with the IS-curve at a different level of output and interest rate. This way the AD-curve, which depicts all combinations of the price level and level of output demanded at which the goods and money sectors are simultaneously in an equilibrium, can be easily derived graphically. It should be made clear that any factor that will affect the slope of either the IS- or the LM-curve will also affect the slope of the AD-curve. Instructors may not want to spend any time showing this either graphically or algebraically, but it is important to distinguish between a movement along the AD-curve, due to a price adjustment and a subsequent change in real balances, and a shift in the AD-curve, due to monetary policy, that is, a change in nominal (and therefore real) money supply). This leads to a good discussion of the effects of monetary and fiscal policy, which is dealt with in more detail in Chapter 12. Instructors, who feel particularly adventuresome, may even want to combine an IS-LM diagram with an AD-AS diagram to show the effects of fiscal and monetary policies. This way it can be made 122 clear that, due to the real balance effect, the fiscal and monetary policy multipliers in an AD-AS model are smaller than those in an IS-LM model. However, such analysis can create even more confusion for students who do not like dealing with graphs. Instructors may therefore prefer to spend only a modest amount of time on the IS-LM framework, using a few simple numerical examples. The effects of economic policies on output, prices, and interest rates can be explained just as well by relying primarily on an AD-AS framework. Answers to Problems in the Textbook Discussion Questions 1. The model in Chapter 10 assumed that both the price level and the interest rate were fixed. But the IS- LM model lets the interest rate fluctuate and determines the combination of output demanded and the interest rate for a fixed price level. It should be noted that while the upward-sloping AD-curve in Chapter 10 (the [C+I+G+NX]-line in the Keynesian cross diagram) assumed that interest rates and prices were fixed, the downward-sloping AD-curve that is derived at the end of Chapter 11 from the IS-LM model lets the price level fluctuate and describes all combinations of the price level and the level of output demanded at which the goods and money sector simultaneously are in equilibrium. 2.a. If the expenditure multiplier () 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 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 causes the [C+I+G+NX]-line to shift up by a larger amount and, given a certain size of the expenditure multiplier , this will increase equilibrium income by a larger amount. As a result, the IS-curve will become flatter. 2.b. Monetary policy changes affect interest rates and this leads to a change in intended spending, which is reflected in a change in income. In 2.a. it was explained that a steep IS-curve means either that the multiplier is small or that desired spending is not very interest sensitive. Therefore, an increase in money supply will reduce interest rates. However, this does not result in a large increase in aggregate demand if spending is very interest insensitive. Similarly, if the multiplier is small, then any change in spending will not affect output significantly. Therefore, the steeper the IS-curve, the weaker the effect of monetary policy changes on equilibrium output. 3. Assume that money supply is 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 bring the money sector back to equilibrium. But if money demand is very interest insensitive, then 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 (due to an increase in income) has to be offset by a decrease in the quantity of money demanded (due to an increase in the interest rate) to keep the money sector in equilibrium. But if money demand becomes more income sensitive, a smaller change in income is required for any specific change in the interest rate to keep the money sector in equilibrium. Therefore, the LM-curve becomes steeper as money demand becomes more income sensitive. 123 4.a. A horizontal LM-curve implies that the public is willing to hold whatever money is supplied at any given interest rate. Therefore, changes in income will not affect the equilibrium interest rate in the money sector. But if the interest rate is fixed, we are back to the analysis of the simple Keynesian model used in Chapter 9. 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 level of investment spending is no longer negatively affected by rising interest rates, that is, there is no crowding-out effect. In terms of Figure 10-3, the interest rate not longer serves as the link between the goods and assets 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 Bank of Canada is prepared to peg the interest rate at a certain level, then any change in income will 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 will never change. 5. From the material presented in the text we know that when intended spending becomes more interest sensitive, then the IS-curve becomes flatter. Now assume that an increase in the interest rate stimulates saving and therefore reduces the level of consumption. 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 will now shift down further than previously and the level of equilibrium income will decrease more than before. In other words, the IS- curve has become flatter. This can also be shown algebraically, since we can now write the consumption function as follows: C = C* + 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 slope of the IS-curve has been reduced from (1 - c)/b to (1 - c)/(b + g). 124 6. In the IS-LM model, a simultaneous decline in interest rates and income can only be caused by a shift of the IS-curve to the left. This shift in the IS-curve could 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 and, since consumer confidence was very low, people were not expected to increase their level of spending. In the IS-LM diagram 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 Application Questions: 1.a. Each point on the IS-curve represents an equilibrium in the expenditure sector. Therefore the IS-curve can be derived by setting 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. 1.b. The IS-curve shows all combinations of the interest rate and the level of output such that the expenditure sector (the goods market) is in equilibrium, that is, intended spending is equal to actual output. 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 to the point where 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. 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 market clears. In other words, the LM- curve is upward sloping. 125 1.e. The level of income (Y) and the interest rate (i) at the equilibrium 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 = 2.5. This multiplier is derived in the same way as in Chapter 9. But now intended spending also depends on the interest rate, so we no longer have Y = Ao, but rather Y = (Ao - bi) = (1/[1 - c + ct])(Ao - bi) ==> Y = (2.5)(1,700 - 50i) = 4,250 - 125i. 2.b.This can be answered most easily with a numerical example. Assume that government purchases increase by G = 300. The IS-curve shifts parallel to the right by ==> IS = (2.5)(300) = 750. Therefore 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 ==> Y = 500 When interest rates are assumed to be constant, the size of the multiplier is equal to = 2.5, that is, (Y)/(G) = 750/300 = 2.5. But when interest rates are allowed to vary, the size of the multiplier is reduced to 1 = (Y)/(G) = 500/300 = 1.67. 126 2.c. Since an increase in government purchases by G = 300 causes a change in the interest rate of 2 percentage points, government spending has to change by G = 150 to increase the interest rate by 1 percentage point. 2.d. The simple multiplier 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 the interest rate. The increase in the interest rate crowds out some investment spending and this has a dampening effect on income. The multiplier effect in 2.b. is therefore smaller than the multiplier effect in 2.a. 3.a. The LM-curve becomes flatter as money supply becomes interest sensitive. Any increase in income will lead to an increase in money demand, which will drive up the interest rate. But the higher interest rates will not only reduce the demand for money but also increase the supply of money. Thus a smaller increase in the interest rate than before is required to bring the money sector back into equilibrium. This means that the LM-curve is flatter. 3.b. If the central bank believes that private spending is very interest sensitive, it is more likely to pursue policies that are intended to keep interest rates from fluctuating. If the central bank believes that most economic disturbances result from changes in money demand (causing the LM-curve to shift), then it is more likely to increase money supply in response to higher interest rates. In other words, if the Bank of Canada accommodates an increase in money demand by increasing money supply, then neither income nor the interest rate will be affected. On the other hand, a disturbance can also come from the expenditure sector, in which case the IS-curve will shift. If interest rates increase due to higher desired spending, then the rise in interest rates can again be kept in check if the Bank of Canada responds by increasing money supply. This time, however, the overall increase in the level of output demanded will now be even larger due the expansionary monetary policy and this may not be desirable. 4.a. An increase in the income tax rate (t) will reduce the size of the expenditure multiplier (). But as the multiplier becomes smaller, the IS-curve becomes steeper. As we can see from the equation for the IS-curve, this is not a parallel shift but rather a rotation around the vertical intercept. Y = (Ao - bi) = [1/(1 - c + ct)](Ao - bi) ==> i = (1/b)Ao - (/b)Y = (1/b)Ao - (1/b)[1 - c + ct]Y 4.b. If the IS-curve shifts to the left and becomes steeper, the equilibrium income level will decrease. A higher tax rate reduces private spending and this will lower national income. 4.c. When the income tax rate is increased, the equilibrium interest rate will also decrease. The adjustment to the new equilibrium can be expressed as follows (see graph on the next page): t up ==> C down ==> Y down ==> md down ==> i down ==> I up ==> Y up. Effect: Y and i 127 IS1 i ISo LM i1 i2 0 Y2 Y1 Y 5.a. If money demand is less interest sensitive, then the LM-curve is steeper and monetary policy changes affect equilibrium income to a larger degree. If money supply is assumed to be fixed, 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; however, in the case of a more interest-sensitive money demand (a flatter LM-curve), the change in Y and i will be smaller. (M/P) up ==> i down ==> I up ==> Y up ==> md up ==> i up Effect: Y and i Section 10-5 derives the equation for the LM-curve and the equation for the monetary policy multiplier as i = (1/h)[kY - (M/P)] and (Y)/(M/P) = (b/h) 128 respectively. 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. 5.b. An increase in money supply drives interest rates down. This decrease in interest rates will stimulate intended spending and thus income. If money demand becomes less interest sensitive, a larger increase in income is required to bring the money sector into equilibrium. But this implies that the overall decrease in the interest rate has to be larger, given that the interest sensitivity of spending has not changed. 6. The price adjustment, that is, the movement along the AD-curve, can be explained in the following way: With nominal money supply (M) fixed, real money balances (M/P) will decrease as the price level (P) increases. There is an excess demand for money and interest rates will rise. This will lead to a decrease in investment spending and thus the level of output demanded will decrease. In other words, the LM-curve will shift to the left as real money balances decrease. 7. In the classical case, the AS-curve is vertical. Therefore, any increase in aggregate demand due to expansionary monetary policy will, in the long run, not lead to any increase in output but simply lead to an increase in the price level. An increase in money supply will first shift the LM-curve to the right. This implies a shift of the AD-curve to the right. Therefore we have excess demand for goods and services and prices will begin to rise. But as the price level rises, real money balances will begin to fall again, eventually returning to their original level. Therefore, the shift of the LM-curve to the right due to the expansionary monetary policy and the resulting shift of the AD-curve will be exactly offset by a shift of the LM-curve to the left and a movement along the AD-curve to the new long-run equilibrium due to the price adjustment. At this new long-run equilibrium, the level of output and interest rates will not have changed while the price level will have changed proportionally to the nominal money supply, leaving real money balances unchanged. In other words, money is neutral in the long run (the classical case). 8.a. An increase in the demand for money will shift the LM-curve to the left, raising the interest rate and lowering the level of output demanded. As a result, the AD-curve will also shift to the left. In the Keynesian case, the price level is assumed to be fixed, that is, the AS-curve is horizontal. In this case, the decrease in income in the AD-AS diagram is equivalent to the decrease in income in the IS-LM diagram, since there is no price adjustment, that is, the real balance effect does not come into play. 8.b. An increase in the demand for money will shift the LM-curve to the left, raising the interest rate and lowering the level of output demanded. As a result, the AD-curve will also shift to the left. In the classical case, the level of output will not change, since the AS-curve is vertical. In this case, the shift in the AD-curve will simply be reflected in a price decrease, but the level of output will remain unchanged. The real balance effect causes the LM-curve to shift back to its original level, since the price decrease causes an increase in real money balances. 129 Additional Problems: 1. True or false? Explain your answer. “A decrease in the marginal propensity to save implies that the IS-curve will become steeper.” False. A decrease in the marginal propensity to save (s = 1 - c) is equivalent to an increase in the marginal propensity to consume (c), which, in turn, implies an increase in the expenditure multiplier (). But with a larger expenditure multiplier, any increase in investment spending due to a decrease in the interest rate will lead to a larger increase in income. Therefore the IS-curve will become flatter and not steeper. 2. True or false? Explain your answer. “If the central bank keeps the supply of money constant, then the money supply curve is vertical, which implies a vertical LM-curve.” False. Equilibrium in the money sector implies that real money supply is equal to real money demand, that is, ms = M/P = md(i,Y). This implies that any increase in income (Y) will increase the demand for money. To bring the money sector back into equilibrium, interest rates (i) have to rise simultaneously to bring the quantity of money demanded back to the original level (equal to the fixed supply of money). Therefore, to keep the money sector in equilibrium, an increase in income must always be associated with an increase in the interest rate and the LM-curve must be upward sloping. 3. "Restrictive monetary policy reduces consumption and investment." Comment on this statement. A reduction in money supply raises interest rates, which will, in turn, have a negative effect on the level of investment spending. The level of consumption may also decrease as it becomes more costly to finance expenditures by borrowing money. But even if it is assumed that consumption is not affected by changes in the interest rate, consumption will still decrease since restrictive monetary policy will reduce national income and therefore private spending. 4. "If government spending is increased, money demand will increase." Comment. A change in government spending directly affects the expenditure sector and therefore the IS-curve. But in an IS-LM framework, the money sector is also affected indirectly. An increase in the level of government spending will shift the IS-curve to the right, leading to an increase in income. But the increase in income will lead to an increase in money demand, so the interest rate will have to increase in order to lower the quantity of money demanded and to bring the money sector back into equilibrium. Overall, no change in money demand can occur, since equilibrium in the money sector requires that ms = M/P = md, that is, money supply has to be equal to money demand, and money supply is assumed to be fixed. 130 5. "An increase in autonomous investment reduces the interest rate and therefore the money sector will no longer be in equilibrium." Comment on this statement. An increase in autonomous investment shifts the IS-curve to the right. The increase in income leads to an increase in the demand for money, which means that interest rates increase. The increase in interest rates then reduces the quantity of money demanded again to bring the money market back to equilibrium. 6. "A monetary expansion leaves the budget surplus unaffected." Comment on this statement. Expansionary monetary policy, that is, an increase in money supply, will lower interest rates (the LM- curve will shift to the right). Lower interest rates will lead to an increase in investment spending and the economy will therefore be stimulated. But a higher level of national income increases the government’s tax revenues and therefore the budget surplus will increase. 7. "Restrictive monetary policy implies lower tax revenues and, therefore, leads to an increase in the budget deficit." Comment on this statement. A decrease in money supply will shift the LM-curve to the left. This will lead to an increase in the interest rate, which will lead to a reduction in spending and thus national income. But as income decreases, so does income tax revenue. Therefore, the budget deficit will increase because of the change in its cyclical component. 8. “If the demand for money becomes more sensitive to changes in income, then the LM-curve becomes flatter.” Comment on this statement. 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 (due to an increase in income) has to be offset by a decrease in the quantity of money demanded (due to an increase in the interest rate) to keep the money sector in equilibrium. But if money demand becomes more income sensitive, a smaller change in income is required for any specific change in the interest rate to keep the money sector in equilibrium. Therefore, the LM-curve becomes steeper (and not flatter) as money demand becomes more sensitive to changes in income. 9. “A decrease in the income tax rate will increase the demand for money, shifting the LM-curve to the right.” Comment on this statement. A decrease in the income tax rate (t) will increase the expenditure multiplier (). But with a larger expenditure multiplier, any increase in investment spending due to a decrease in the interest rate will lead to a larger increase in income. Since fiscal policy affects the expenditure sector, the IS-curve (not the LM- curve) will shift. The IS-curve will become flatter and shift to the right. This will lead to a new equilibrium at a higher level of income (Y) and a higher interest rate (i). But money supply is fixed and the LM-curve remains unaffected by fiscal policy. Therefore, at the new equilibrium (the intersection of the new IS-curve with the old LM-curve) the demand for money will not have changed, since the money sector has to be in an equilibrium at ms = md(i,Y). 131 10. “If the demand for money becomes more insensitive to changes in the interest rate, equilibrium in the money sector will have to be restored mostly through changes in income. This implies a flat LM-curve.” Comment on this statement. Any increase in income will increase money demand and this will drive the interest rate up. Therefore, the quantity of money balances demanded will decline again until the money sector is back in equilibrium. But if money demand is very interest insensitive, then a larger increase in the interest rate is needed to reach a new equilibrium in the money sector. This means that the LM-curve is steep and not flat. 11. Assume the following IS-LM model: Expenditure Sector Money Sector Sp = C + I + G + NX M = 700 C = 100 + (4/5)YD P =2 YD = Y - TA md = (1/3)Y + 200 - 10i TA = (1/4)Y I = 300 - 20i G = 120 NX = -20 (a) Derive the equilibrium values of consumption (C) and money demand (md). (b) How much of investment (I) will be crowded out if the government increases its purchases by G = 160 and nominal money supply (M) remains unchanged? (c) By how much will the equilibrium level of income (Y) and the interest rate (i) change, if nominal money supply is also increased to M' = 1,100? a. Sp = 100 + (4/5)[Y - (1/4)Y] + 300 - 20i + 120 - 20 = 500 + (4/5)(3/4)Y – 20i = 500 + (3/5)Y - 20i From Y = Sp ==> Y = 500 + (3/5)Y - 20i ==> (2/5)Y = 500 - 20i ==> Y = (2.5)(500 - 20i) ==> Y = 1,250 - 50i IS-curve From M/P = md ==> 700/2 = (1/3)Y + 200 - 10i ==> (1/3)Y = 150 + 10i ==> Y = 3(150 + 10i) ==> Y = 450 + 30i LM-curve IS = LM ==> 1,250 - 50i = 450 + 30i ==> 800 = 80i ==> i = 10 ==> Y = 1,250 - 50*10 ==> Y = 750 C = 100 + (4/5)(3/4)750 = 100 + (3/5)750 ==> C = 550 ms = M/P = 700/2 = 350 = md Check: md = (1/3)750 + 200 - 10*10 = 350 132 i 25 ISo LMo 10 0 450 750 1,250 Y b. IS = (2.5)160 = 400 ==> IS' = 1,650 - 50i IS' = LM ==> 1,650 - 50i = 450 + 30i ==> 1,200 = 80i ==> i = 15 ==> Y = 1,650 - 50*15 ==> Y = 900 Since i = + 5 ==> I = - 20*5 ==> I = - 100 Check: Sp = G + I = 160 – 100 = 60 ==> Y = 1(Sp) = 2.5*60 =150 i IS1 33 25 LMo 15 10 0 450 750 900 1,250 1,650 Y c. From M'/P = md ==> 1,100/2 = (1/3)Y + 200 - 20i ==> (1/3)Y = 350 - 20i ==> Y = 3(350 - 20i) ==> Y = 1,050 + 30i IS1 = LM1 ==> 1,650 - 50i = 1,050 + 30i ==> 600 = 80i ==> i = 7.5 ==> Y = 1,650 - 50(7.5) = 1,275. ==> i = - 7.5 and Y = 375 as compared to (b). 133 i 25 IS1 LMo LM1 10 7.5 0 450 900 1,050 1,275 1,650 Y 12. Assume the money sector can be described by these equations: M/P = 400 and md = (1/4)Y - 10i. In the expenditure sector only investment spending (I) is affected by the interest rate (i), and the equation of the IS-curve is: Y = 2,000 - 40i. (a) If the size of the expenditure multiplier is = 2, show the effect of an increase in government purchases by G = 200 on income and the interest rate. (b) Can you determine how much of investment is crowded out as a result of this increase in government spending? (c) If the money demand equation were changed to md = (1/4)Y, how would your answers in (a) and (b) change? a. From M/P = md ==> 400 = (1/4)Y - 10i ==> Y = 1,600 + 40i LM-curve From IS = LM ==> 2,000 - 40i = 1,600 + 40i ==> 80i = 400 ==> i = 5 ==> Y = 2,000 - 40*5 ==> Y = 1,800 IS = 2*200 = 400 ==> IS' = 2,400 - 40i IS' = LM ==> 2,400 - 40i = 1,600 + 40i ==> 80i = 800 ==> i = 10 ==> Y = 1,600 + 40*10 ==> Y = 2,000 Therefore i = + 5 and Y = + 200 b. Since the size of the expenditure multiplier is = 2 but income only goes up by Y = 200, the fiscal policy multiplier in the IS-LM model is 1= 1. But this means that the level of investment has been reduced by 100, that is, I = -100. This can be seen by restating the IS-curve as follows: Y = 2,000 - 40i = Y = 2(1,000 - 20i) Since government purchases are changed by G = 200 ==> Y = 2(1,200 - 20i), which means that the IS-curve shifts by IS = 2*200 = 400. But the increase in income is actually only Y = 200. This implies that investment changes by I = -100. Investment is of the form I = Io – 20i; however, since the interest rate went up by i = 5, investment changes by I = - 20*5 = - 100. 134 From Y = (Sp) ==> 200 = 2(Sp) ==> Sp = 100 But since Sp = G + I ==> 100 = 200 + I ==> I = - 100 c. If md = (1/4)Y, then we have the classical case, that is, a vertical LM-curve. In this case, fiscal expansion will not change income at all. This occurs since the increase in G will be offset by a decrease in I of equal magnitude due to an increase in the interest rate. (M/P) = md ==> 400 = (1/4)Y ==> Y = 1,600 LM-curve IS = LM ==> 2,000 - 40i = 1,600 ==> 40i = 400 ==> i = 10 ==> Y = 1,600 IS' = LM ==> 2,400 - 40i = 1,600 ==> 40i = 800 ==> i = 20 ==> Y = 1,600 ==> I = - 200 13. Assume money demand (md) and money supply (ms) are defined as: md = (1/4)Y + 400 - 15i and ms = 600, and intended spending is of the form: Sp = C + I + G + NX = 400 + (3/4)Y - 10i. Calculate the equilibrium levels of Y and i, and indicate by how much the Bank of Canada would have to change money supply to keep interest rates constant if the government increased its spending by G = 50. Show your solutions graphically and mathematically. ms = md ==> 600 = (1/4)Y + 400 - 15i ==> (1/4)Y = 200 + 15i ==> Y = 4(200 + 15i) ==> Y = 800 + 60i LM-curve Y = C + I + G + NX ==> Y = 400 + (3/4)Y - 10i ==> (1/4)Y = 400 - 10i ==> Y = 4(400 - 10i) ==> Y = 1,600 - 40i IS-curve From IS = LM ==> 1,600 - 40i = 800 + 60i ==> 100i = 800 ==> i = 8 ==> Y = 1,280 If government spending is increased by G = 50, the IS-curve will shift to the right) by (IS) = 4*50 = 200. If the Bank of Canada wants to keep the interest rate constant, money supply has to be increased in a way that shifts the LM-curve to the right by exactly the same amount as the IS-curve, that is, (LM) = 200. From Y = 2(200 + 15i) ==> (Y) = 2(ms) ==> 200 = 2(ms) ==> (ms) = 100, so money supply has to be increased by 100. Check: IS' = LM": 1,800 - 40i = 1,000 + 60i ==> 800 = 100i ==> i = 8 Y = 1,480 135 i 45 IS1 40 ISo LMo LM1 8 0 800 1000 1280 1480 1600 1800 Y 14. Assume the equation for the IS-curve is Y = 1,200 – 40i, and the equation for the LM-curve is Y = 400 + 40i. (a) Determine the equilibrium value of Y and i. (b) If this is a simple model without income taxes, by how much will these values change if the government increases its expenditures by G = 400, financed by an equal increase in lump sum taxes (TAo = 400)? a. From IS = LM ==> 1,200 - 40i = 400 + 40i ==> 800 = 80i ==> i = 10 ==> Y = 400 + 40*10 ==> Y = 800 b. According to the balanced budget theorem, the IS-curve will shift horizontally by the increase in government purchases, that is, IS = G = TAo = 400. Thus the new IS-curve is of the form: Y = 1,600 - 40i. From IS' = LM ==> 1,600 - 40i = 400 + 40i ==> 1,200 = 80i ==> i = 15 ==> Y = 400 + 40*15 ==> Y = 1,000 15. Assume you have the following information about a macro model: Expenditure sector: Money sector: S = - 200 + (1/5)YD ms = 400 TA = (1/8)Y - 40 md = (1/4)Y + 100 - 5i TR = 60 I = 300 – 10i G = 70 NX = 150 - (1/5)Y Calculate the equilibrium values of investment (I), money demand (md), and net exports (NX). C = YD - S = YD – [-200 + (1/5)Y] = 200 + (4/5)YD 136 Sp = C + I + G + NX = 200 + (4/5)[Y- (1/8)Y + 40 + 60] + 300 - 10i + 70 + 150 - (1/5)Y = 720 + (4/5)(7/8)Y + (4/5)100 - 10i - (1/5)Y = 800 + (1/2)Y - 10i Y = Sp ==> Y = 800 + (1/2)Y - 10i ==> (1/2)Y = 800 - 10i Y = 2(800 - 10i) ==> Y = 1,600 - 20i IS-curve ms = md ==> 400 = (1/4)Y + 100 - 5i ==> (1/4)Y = 300 + 5i ==> Y = 1,200 + 20i LM-curve IS = LM ==> 1,600 - 20i = 1,200 + 20i ==> 40i = 400 ==> i = 10 Y = 1,400 ==> I = 300 - 10*10 = 200 NX = 150 - (1/5)1,400 = - 130 md = ms = 300 16. Assume the following IS-LM model: expenditure sector: money sector: Sp = C + I + G + NX M = 500 C = 110 + (2/3)YD P =1 YD = Y - TA + TR md = (1/2)Y + 400 - 20i TA = (1/4)Y + 20 TR = 80 I = 250 - 5i G = 130 NX = -30 (a) Calculate the equilibrium values of investment (I), real money demand (m d), and tax revenues (TA). (b) How much of investment (I) will be crowded out if the government increases spending by G = 100? a. Sp = C + I + G + NX = 110 + (2/3)(Y - TA + TR) + 250 - 5i + 130 - 30 = 460 + (2/3)[Y - (1/4)Y - 20 + 80] - 5i = 460 + (2/3)(3/4)Y + (2/3)60 - 5i = 500 + (1/2)Y - 5i From Y = Sp ==> Y = 500 + (1/2)Y - 5i ==>Y = 2(500 - 5i) ==> Y = 1,000 - 10i IS-curve From (M/P) = md ==> 500/1 = (1/2)Y + 400 - 20i ==> (1/2)Y = 100 + 20i ==> Y = 200 + 40i LM-curve From IS = LM ==> 1,000 - 10i = 200 + 40i ==> 800 = 50i ==> i = 16 ==> Y = 840 ==> I = 250 - 5*16 = 170 TA = (1/4)840 - 20 = 190 Since (M/P) = md ==> md = 500 Check: md = (1/2)840 + 400 - 20*16 = 500 137 b. If government expenditures are increased by G = 100, then the IS-curve will shift by this change times the multiplier, that is, IS = 2*100 = 200. Therefore, from IS' = LM ==> 1,200 - 10i = 200 + 40i ==>1,000 = 50i ==> i = 20 ==> Y = 1,000 Since i = 4 ==> I = - 4*5 = - 20 Check: I' = 250 - 5*20 = 150 138