CH-11 Aggregate Demand I: Building the IS–LM Model – Macroeconomics : Book Guide

Answers to Textbook Questions and Problems CHAPTER 11 Aggregate Demand I: Building the IS–LM Model Questions for Review 1. The Keynesian-cross model tells us that fiscal policy has a multiplied effect on income. The reason is that according to the consumption function, higher income causes higher consumption. For example, an increase in government purchases of ΔG raises expenditure and, therefore, income by ΔG. This increase in income causes consumption to rise by MPC  ΔG, where MPC is the marginal propensity to consume. This increase in consumption raises expenditure and income even further. This feedback from consumption to income continues indefinitely. Therefore, in the Keynesian-cross model, increasing government spending by one dollar causes an increase in income that is greater than one dollar: it increases by 1/(1 – MPC). 2. The theory of liquidity preference explains how the supply and demand for real money balances determine the interest rate. A simple version of this theory assumes that there is a fixed supply of money, which the Fed chooses. The price level P is also fixed in this model, so that the supply of real balances is fixed. The demand for real money balances depends on the interest rate, which is the opportunity cost of holding money. At a high interest rate, people hold less money because the opportunity cost is high. By holding money, they forgo the interest on interest-bearing deposits. In contrast, at a low interest rate, people hold more money because the opportunity cost is low. Figure 11-1 illustrates the supply and demand for real money balances. Based on this theory of liquidity preference, the interest rate adjusts to equilibrate the supply and demand for real money balances. Why does an increase in the money supply lower the interest rate? Consider what happens when the Fed increases the money supply from M1 to M2. Because the price level P is fixed, this increase in the money supply shifts the supply of real money balances M/P to the right, as in Figure 11-2. The interest rate must adjust to equilibrate supply and demand. At the old interest rate r1, supply exceeds demand. People holding the excess supply of money try to convert some of it into interestbearing bank deposits or bonds. Banks and bond issuers, who prefer to pay lower interest rates, respond to this excess supply of money by lowering the interest rate. The interest rate falls until a new equilibrium is reached at r2. 3. The IS curve summarizes the relationship between the interest rate and the level of income that arises from equilibrium in the market for goods and services. Investment is negatively related to the interest rate. As illustrated in Figure 11-3, if the interest rate rises from r1 to r2, the level of planned investment falls from I1 to I2. The Keynesian cross tells us that a reduction in planned investment shifts the expenditure function downward and reduces national income, as in Figure 11-4(A). Thus, as shown in Figure 11-4(B), a higher interest rate results in a lower level of national income: the IS curve slopes downward. 4. The LM curve summarizes the relationship between the level of income and the interest rate that arises from equilibrium in the market for real money balances. It tells us the interest rate that equilibrates the money market for any given level of income. The theory of liquidity preference explains why the LM curve slopes upward. This theory assumes that the demand for real money balances L(r, Y) depends negatively on the interest rate (because the interest rate is the opportunity cost of holding money) and positively on the level of income. The price level is fixed in the short run, so the Fed determines the fixed supply of real money balances M/P. As illustrated in Figure 11-5(A), the interest rate equilibrates the supply and demand for real money balances for a given level of income. Now consider what happens to the interest rate when the level of income increases from Y1 to Y2. The increase in income shifts the money demand curve upward. At the old interest rate r1, the demand for real money balances now exceeds the supply. The interest rate must rise to equilibrate supply and demand. Therefore, as shown in Figure 11-5(B), a higher level of income leads to a higher interest rate: The LM curve slopes upward. Problems and Applications 1. a. The Keynesian cross illustrates an economy’s planned expenditure function, PE = C(Y – T) + I + G, and the equilibrium condition that actual expenditure equals planned expenditure, Y = PE, as shown in Figure 11-6. An increase in government purchases from G1 to G2 shifts the planned expenditure function upward. The new equilibrium is at point B. The change in equilibrium GDP, Y, equals the product of the government-purchases multiplier and the change in government spending: ΔY = [1/(1 – MPC)]ΔG. Because we know that the marginal propensity to consume MPC is less than one, this expression tells us that a one-dollar increase in G leads to an increase in Y that is greater than one dollar. b. An increase in taxes T reduces disposable income Y – T by ΔT and, therefore, reduces consumption by MPC  ΔT. For any given level of income Y, planned expenditure falls. In the Keynesian cross, the tax increase shifts the planned-expenditure function down by MPC  ΔT, as in Figure 11-7. The amount by which equilibrium GDP falls is given by the product of the tax multiplier and the increase in taxes: ΔY = [–MPC/(1 – MPC)]ΔT. c. We can calculate the effect of an equal increase in government expenditure and taxes by adding the two multiplier effects that we used in parts (a) and (b): ΔY = {[1/(1 – MPC)]ΔG} – {[MPC/(1 – MPC)]ΔT}. Government Tax Spending Multiplier Multiplier Because government purchases and taxes increase by the same amount, we know that ΔG = ΔT. Therefore, we can rewrite the above equation as ΔY = {[1/(1 – MPC)] – [MPC/(1 – MPC)]}ΔG = ΔG. This expression tells us that an equal increase in government purchases and taxes increases Y by the amount that G increases. That is, the balanced-budget multiplier is exactly 1. 2. a. Total planned expenditure is PE = C(Y – T) + I + G. Plugging in the consumption function and the values for investment I, government purchases G, and taxes T given in the question, total planned expenditure PE is PE = 120 + 0.80(Y – 400) + 200 + 400 = 0.80Y + 400. This equation is graphed in Figure 11-8. Figure 11-8 b. To find the equilibrium level of income, combine the planned-expenditure equation derived in part (a) with the equilibrium condition Y = PE: Y = 0.80Y + 400 Y = 2,000. The equilibrium level of income is 2,000, as indicated in Figure 11-8. c. If government purchases increase to 420, then planned expenditure changes to PE = 0.80Y + 420. Equilibrium income increases to Y = 2,100. Therefore, an increase in government purchases of 20 (i.e., 420 – 400 = 20) increases income by 100. This is what we expect to find, because the formula for the government-purchases multiplier is 1/(1 – MPC), the MPC is 0.80, and the government-purchases multiplier therefore has a numerical value of 5. d. An income level of 2,400 represents an increase of 400 over the original level of income. The government-purchases multiplier is 1/(1 – MPC): the MPC in this example equals 0.80, so the government-purchases multiplier is 5. This means that government purchases must increase by 80 (to a level of 480) for income to increase by 400. e. An income level of 2,400 represents an increase of 400 over the original level of income. The tax multiplier is –MPC/(1 – MPC): the MPC in this example equals 0.80, so the tax multiplier is 4. This means that taxes must decrease by 100 (to a level of 300) for income to increase by 400. 3. a. When taxes do not depend on income, a one-dollar increase in income means that disposable income increases by one dollar. Consumption increases by the marginal propensity to consume MPC. When taxes do depend on income, a one-dollar increase in income means that disposable income increases by only (1 – t) dollars. Consumption increases by the product of the MPC and the change in disposable income, or (1 – t)MPC. This is less than the MPC. The key point is that disposable income changes by less than total income, so the effect on consumption is smaller. b. When taxes are fixed, we know that ΔY/ΔG = 1/(1 – MPC). We found this by considering an increase in government purchases of ΔG; the initial effect of this change is to increase income by ΔG. This in turn increases consumption by an amount equal to the marginal propensity to consume times the change in income, MPC  ΔG. This increase in consumption raises expenditure and income even further. The process continues indefinitely, and we derive the multiplier above. When taxes depend on income, we know that the increase of ΔG increases total income by ΔG; disposable income, however, increases by only (1 – t)ΔG . Consumption then increases by an amount (1 – t) MPC  ΔG. Expenditure and income increase by this amount, which in turn causes consumption to increase even more. The process continues, and the total change in output is ΔY = ΔG {1 + (1 – t)MPC + [(1 – t)MPC]2 + [(1 – t)MPC]3 + ....} = ΔG {1/[1 – (1 – t)MPC]}. Thus, the government-purchases multiplier becomes 1/[1 – (1 – t)MPC] rather than 1/(1 – MPC). This means a much smaller multiplier. For example, if the marginal propensity to consume MPC is 3/4 and the tax rate t is 1/3, then the multiplier falls from 1/(1 – 3/4), or 4, to 1/[1 – (1 – 1/3)(3/4)], or 2. c. In this chapter, we derived the IS curve algebraically and used it to gain insight into the relationship between the interest rate and output. To determine how this tax system alters the slope of the IS curve, we can derive the IS curve for the case in which taxes depend on income. Begin with the national income accounts identity: Y = C + I + G. The consumption function is C = a + b(Y – T – tY). Note that in this consumption function taxes are a function of income. The investment function is the same as in the chapter: I = c – dr. Substitute the consumption and investment functions into the national income accounts identity to obtain Y = [a + b(Y – T – tY)] + c – dr + G. Solving for r: r = a-bT +c+G +YØŒ b(1-t)-1øœ . d Œº d œß The slope of the IS curve is therefore Dr b(1-t)-1 = . Dy d Recall that t is a number that is less than 1. As t becomes a bigger number, the slope of the IS curve increases in absolute value terms and the curve becomes steeper. Suppose, for example, that b is 0.80, t is 0.1, and d is 0.5. The slope of the IS curve is –0.56. If the tax rate increases to 0.2, then the slope becomes –0.72. Intuitively, if the tax rate is higher, then any given reduction in the interest rate has a smaller effect on real output Y because the multiplier will be smaller. 4. a. If society becomes more thrifty—meaning that for any given level of income people save more and consume less—then the planned-expenditure function shifts downward, as in Figure 11-9 (note that C2 10% ▪ Oct 1979: Fed Chairman Paul Volcker announces that monetary policy would aim to reduce inflation ▪ Aug 1979–April 1980: Fed reduces M/P 8.0% ▪ Jan 1983: π = 3.7% How do you think this policy change would affect nominal interest rates? The LM curve Now let’s put Y back into the money demand function: (MP)d = LrY( , ) The LM curve is a graph of all combinations of r and Y that equate the supply and demand for real money balances. The equation for the LM curve is: MP LrY= ( , ) Deriving the LM curve (a) The market for Why the LM curve is upward sloping ▪ An increase in income raises money demand. ▪ Since the supply of real balances is fixed, there is now excess demand in the money market at the initial interest rate. ▪ The interest rate must rise to restore equilibrium in the money market. How ΔM shifts the LM curve (a) The market for The short-run equilibrium The short-run equilibrium is r the combination of r and Y that simultaneously satisfies the equilibrium conditions in the goods & money markets: Y CY T Ir G= ( − +) ( )+ MP LrY= ( , ) The Big Picture Preview of Chapter 12 In Chapter 12, we will ▪ use the IS-LM model to analyze the impact of policies and shocks. ▪ learn how the aggregate demand curve comes from IS-LM. ▪ use the IS-LM and AD-AS models together to analyze the short-run and long-run effects of shocks. ▪ use our models to learn about the Great Depression. 1. Keynesian cross ▪ basic model of income determination ▪ takes fiscal policy & investment as exogenous ▪ fiscal policy has a multiplier effect on income 2. IS curve ▪ comes from Keynesian cross when planned investment depends negatively on interest rate ▪ shows all combinations of r and Y that equate planned expenditure with actual expenditure on goods & services 3. Theory of liquidity preference ▪ basic model of interest rate determination ▪ takes money supply & price level as exogenous ▪ an increase in the money supply lowers the interest rate 4. LM curve ▪ comes from liquidity preference theory when money demand depends positively on income ▪ shows all combinations of r and Y that equate demand for real money balances with supply 5. IS-LM model ▪ Intersection of IS and LM curves shows the unique point (Y, r ) that satisfies equilibrium in both the goods and money markets. Solution Manual for Macroeconomics Gregory N. Mankiw 9781464182891, 9781319106058