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Transcript
7. IS-LM and Aggregate Demand A fundamental idea embodied in the income-expenditure accounting identity is that one person’s spending becomes another person’s income. The simple circular flow diagram presented in Figure 3.1 clearly illustrates this idea for a closed economy without government. Re-examining that figure, notice that expenditure not only becomes income, but the reverse is also true: income becomes expenditure. This interaction confounds cause and effect. Do expenditures generate income? Or, does income generate expenditures? To examine this issue, macroeconomists have developed the concept of aggregate demand, which is the total willingness of buyers to buy goods and services over a given period of time. Aggregate demand is comparable to aggregate expenditure. However, whereas aggregate expenditure is an accounting concept measuring the actual expenditure on goods and services, aggregate demand is a modeling concept representing the buying plans of buyers. One would expect buying plans to be contingent upon a variety of factors. The central question of interest here is, “What determines buyers’ willingness to buy?” Or, using economic jargon, “What determines aggregate demand?” This is an important question since producers cannot sell all of their output if the willingness to buy is insufficient. 7.1 Origin of the IS-LM Model In his classic 1936 book, The General Theory of Employment, Interest, and Money, John Maynard Keynes provided a critique of what he called the classical theory and offered an alternative. Writing in the Great Depression, Keynes’ focus was on understanding what determines the economy’s employment level. The concept of aggregate demand is central to Keynes’ theory. Keynes presented few of his ideas using mathematical models. Preferring verbal discourse, he specifically criticized mathematical formalization, saying “Too large a portion of recent ‘mathematical’ economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.” Ironically, professional economists today use mathematical modeling much more than economists did in Keynes’ day, though many economists today would whole-heartedly agree with this quote by Keynes. Mathematical modeling has become popular because it allows the assumptions of the analysis to be precisely stated. While words allow for subtlety and complexity that is difficult to capture with mathematics, they also allow for more varied interpretations, which can lead to confusion. For example, although Keynes was an organized thinker and writer, economists today still hotly debate the meaning of his words. 89 Aggregate Demand Chapter 7: IS-LM and Aggregate Demand IS Curve LM Curve it allows the model builder to more carefully identify and study cause and effect relationships. The interdependencies and complexities that Keynes spoke of can make non-mathematical analysis exceedingly difficult. Mathematical methods exist which can simplify complicated relationships. it allows econometric analysis. To do science, theories must be tested against the facts. Econometric models are mathematical models that allow this testing. John Hicks presented his view of Keynes’ work relative to Classical thought in his famous 1937 essay, “Mr. Keynes and the ‘Classics’.” There Hicks introduced what is now know as the IS-LM model. (For Hicks it was IS-LL.) In its graphical form, this model includes an IS curve---the real interest rate and output combinations which are consistent with investment being equal to savings. It also includes an LM curve---the real interest rate and output combinations which are consistent with liquidity preference being equal to the money supply. These relationships will now be explained in more detail. 7.2 The IS Curve As a starting point, reconsider equation (2.1), which we reproduce here without the time subscripts: (7.1) Y C I G NX . The left side of (7.1) is the economy’s output level, which national income accountants now measure as gross domestic product. The right side of (7.1) includes the expenditure components of output. To go from accounting to modeling, one can re-interpret (7.1) as an equilibrium condition where plans are actualized. That is, suppose the left side of (7.1) still represents the economy’s actual output level. However, rather than thinking of the right side of (7.1) as the actual expenditure components of output as measured in the national income and product accounts, think of it as aggregate demand as defined above. That is, we represent the total willingness to buy goods and services as the sum of the willingness households to buy consumption goods---consumption demand, the willingness of firms to buy investment goods---investment demand, the willingness of government to buy goods--government purchases, and the net willingness of foreigners to buy goods. Letting AD denote aggregate demand, this implies (7.2) AD C I G NX In order for the plans of buyers and sellers to be simultaneously fulfilled, the equilibrium condition 90 Chapter 7: IS-LM and Aggregate Demand (7.3) Y AD must hold. Note that conditions (7.2) and (7.3) imply (7.1) must hold in equilibrium. Will the economy tend toward an equilibrium? Disequilibrium is certainly possible, if not likely. Keynes, like the Classical economists before him, felt disequilibrium would motivate change that would tend to move the economy toward an equilibrium. However, since Keynes, there has been dispute over what will tend to adjust to bring about the equilibrium. To examine this Keynesian-Classical dispute, we must add more detail to our model. Keynes described an economy where consumption depends upon income. Since we are considering an economy with government, let consumption depend upon disposable income according to the consumption function (7.4) C C(Y T ), 0 C' 1 . The derivative C’ of the consumption function, represents what Keynes called the marginal propensity to consume. Restricting C’ between zero and one indicates that a one dollar increase in disposable income will lead to an increase in consumption but not by the full dollar. The remaining fraction is saved, as can be shown using the income allocation condition from chapter 2, reproduced here without time subscripts as (7.5) Y C S T TF . Consumption Function Marginal Propensity to Consume Eliminating C from (7.5) using (7.4) and solving for S, one obtains the savings function (7.6) S Y T C(Y T ) TF . Equation (7.6) indicates that saving is a function of disposable income Y-T and transfers to foreigners TF. Taking the derivative of the right side of (7.6) with respect to Y-T yields 1-C’, the marginal propensity to save. Combining this knowledge with that obtained above for the marginal propensity to consume, our model indicates that C’ of an additional dollar of disposable income is spent on consumption, while the remaining fraction 1-C’ is saved. Keynes postulated that investment depends upon what he called the marginal efficiency of capital, a measure of the expected return on investment. The marginal efficiency of capital is comparable to the internal rate of return concept used in finance. For a given prospective investment, there are expected cost outlays and expected returns. For the typical investment, costs are typically incurred in more toward the present, while the benefits are mostly received in the future. The internal rate of return is that rate which discounts the benefits to the point where they are equal to the discounted costs. A prospective investment project looks more attractive as its internal rate of return increases. 91 Savings Function Marginal Propensity to Save Marginal Efficiency of Capital Marginal Efficiency of Capital Chapter 7: IS-LM and Aggregate Demand Keynes noted that investment would be induced when the marginal efficiency of capital exceeds the real rate of interest. As such investment occurs, the marginal efficiency of capital decreases, either because of to diminishing returns to capital or because of an increase in the prices of investment goods. Alternatively, investment would not be forthcoming when the real interest rate exceeds the marginal efficiency of capital. In such a situation the marginal efficiency of capital would tend to increase, either because depreciation would make capital increasingly scarce or because the prices of investment goods would decrease. Consequently, Keynes argued that the marginal efficiency of capital will tend to find an equilibrium where it is equal to the real rate of interest. Starting in such an equilibrium, a certeris paribus decrease in the real interest rate would tend to promote investment. Hicks expressed this conclusion in the form of an investment function, such as (7.7) I I (r ) , Investment Function Endogenous Variable Exogenous Variable I ' 0 , where r denotes the real interest rate. Investment occurs in response to an interest rate decrease until the marginal efficiency of capital falls to the new interest rate level. We now have a model of the economy’s circular flow of income and expenditure. It consists of six equations: (7.2), (7.3), (7.4), (7.5), and (7.7). Starting with (7.3) and using substitution and the other conditions, we obtain the following series of relationships Implicit Function (7.8) Y AD Given as (7.3) Y C I G NX Substitution; (7.2) Y C(Y T ) I (r ) G NX Substitution; (7.4) and (7.7) Equation (7.8) is a variant of Hicks’ IS equation. IS equation The capital market is the market for saving. Using the equilibrium condition (7.3) and substitution as follows: Capital Market (7.9) Y AD Given as (7.3) Y C I G NX Substitution; (7.2) S T G NX I TF Substitution; (7.5) 92 Chapter 7: IS-LM and Aggregate Demand Condition (7.9) can be thought of as a capital market equilibrium condition. On the left side of (7.9) is the total saving supplied to the domestic economy, consisting of private saving S, public saving T-G, and foreign saving -NX. On the right side of (7.9) is the domestic demand for saving in the form of domestic investment I and transfers from foreigners -TF. The last series of equations shows that either (7.3) or (7.9) could be used as an equilibrium condition in constructing our model. Hicks used a capital market condition like (7.9). To be precise, Hicks examined an economy without government and without a foreign sector, meaning condition (7.9) reduces to I=S. It is from this condition that the IS curve gets its name. By using condition (7.3), we follow more recent tradition. 7.3 Using the Total Differential to Examine the IS equation Now consider the IS equation (7.8) in more detail. As a single equation, it can determine only one variable, typically called an endogenous variable. The Classical approach is to assume that the real interest rate r is endogenous, adjusting to equate saving and investment whenever they are not equal. The remaining variables are exogenous, determined by outside influences. By treating r as endogenous, we can think of (7.8) as an implicit function determining r in terms of the exogenous variables Y, T, G, and NX. We now show how the total differential can be used to examine how changes in Y, T, G, and NX affect r. Totally differentiating (7.9) gives (7.10) dY C' dY C' dT I ' dr dG d[ NX ] , where the differentials, dY, dT, dG, d[NX], and dr respectively represent the change in output, the change in net taxes, the change in government purchases, the change in net exports, and the change in the real interest rate. Solving (7.10) for dr yields (7.11) dr 1 C' C' 1 1 dY dT dG d[ NX ] . I' I' I' I' Equation (7.11) indicates that the total change in the real interest rate, given by dr, depends upon the change in output dY, the change in net taxes dT, the change in government purchases dG, and the change in net exports d[NX]. The coefficients on the exogenous differentials are called multipliers because a given multiplier indicates how the endogenous variable r changes in response to a one unit r increase in the particular exogenous variable. For example, the multiplier is found Y by setting dT dG d[ NX ] 0 , which yields 93 Chapter 7: IS-LM and Aggregate Demand Multipliers (7.12) r 1 C ' 0. Y I' Notice that a multiplier is a partial derivative found by hold all exogenous variables constant, except the one of interest. The multiplier (7.12) gives the change in r that results from a one unit increase in Y. It also gives the slope of the IS curve, which is typically drawn in (Y,r) space as shown in Figure 7.1 Figure 7.1: The IS Curve r Slope 1 C' I' IS Y Money Demand Transactions Motive Precautionary Motive Speculative Motive The multiplier is negative, indicating that an increase in output leads to a decrease in the real interest rate. The magnitude of the decrease depends upon the size of the marginal propensity to save 1-C’ and the sensitivity of investment to changes in the interest rate I’. The interest rate decreases when output increases because an increase in output generates and increase in income; the increase in income then increases private saving. The increase in saving creates a surplus of saving. To encourage borrowers to borrow the surplus, financial intermediaries decrease the real interest rate level. If the propensity to save 1-C’ is small, then the income increase will not generate much of an increase in saving, meaning interest rates will not have to increase much to eliminate the saving surplus. Alternatively, if I’ is small, meaning investment is insensitive to a change in the interest rate, then a relatively large decrease in the interest rate would be necessary to encourage borrowers to borrow the saving surplus. Given these interpretations of this multiplier, it is apparent that we would like to know the values of C’ and I’ for our economy. r C ' r 1 0, 0 indicate that a reduction in the budget T I ' G I' deficit, either through a tax increase or government purchases decrease, will reduce the real interest rate level . This occurs because a reduction in the budget deficit is equivalent to an increase in public saving. The increase in public saving generates a surplus of saving. Again, to encourage borrowers to borrow the surplus, financial intermediaries decrease the real interest rate level. The multipliers Money Market 94 Chapter 7: IS-LM and Aggregate Demand r 1 0 indicates that a reduction in the trade deficit---which is [ NX ] I' equivalent to an increase the trade surplus or net exports---will increase the real interest rate level. This occurs because a reduction in the trade deficit is equivalent to an decrease in foreign saving. The decrease in foreign saving generates a shortage of saving. To discourage borrowers from borrowing, financial intermediaries increase the real interest rate level. The multiplier 7.4 The LM Curve Hicks argued that Keynes’ innovation was to introduce the notion of “liquidity preferences”---the idea that money demand depends upon the rate of interest. Money demand is the willingness of people to hold money. The Classical approach is to assume that people hold money to make transactions. Because people with higher incomes make more transactions, the Classical approach is to assume money demand L depends upon the income level Y : (7.13) L L(Y ) , L' 0 . In addition to the transactions motive, Keynes spoke of a precautionary motive---hold money for unexpected circumstances---and a speculative motive---hold money in anticipation of a bargain purchase. In any case, the interest rate represents the opportunity cost of holding money. Thus, Keynes argued that an increase in the interest rate would reduce the preference for liquidity; i.e. reduce money demand. Adding the real interest rate i to the money demand function (7.13), one obtains (7.14) L L(Y ,r ) , LY 0 , Lr 0 . While Keynes and Hicks thought of money demand in nominal terms, here we follow the approach that has become more standard: Define L as real money demand and P as the price of output so that PL represents nominal money demand. Letting M denote the nominal money supply, the money market is in equilibrium when (7.15) PL M . Substituting the money demand function (7.14) into (7.15), one obtains Hicks’ LM equation: (7.16) PL(Y ,r ) M . As Hicks explained, Keynes distinguished himself from the Classical perspective by assuming (7.16) determines r. Most significantly, the rate of interest depends upon the quantity of money. 95 LM Equation Chapter 7: IS-LM and Aggregate Demand However, Hicks introduced the IS-LM diagram specifically to show that the Keynesian perspective on the money market is not really much different from the Classical perspective once the money market model---represented by the LM equation--is combined with the capital market model---represented by the IS equation. We will consider that framework momentarily. However, we first examine the shape of the LM curve, something Hicks described as “the most important thing in Keynes’ book.” Totally differentiating (7.16), one obtains (7.17) PLY dY PLr dr LdP dM . Solving for dr, one obtains (7.18) dr 1 L L dM Y dY dP . PLr Lr Lr r L Y 0 gives the slope of the LM curve. The slope of the LM Y Lr curve drawn in Figure 7.2 is not constant but rather increases at an increasing rate. This is the shape Hicks credited to Keynes and deemed so important. Mathematically, this shape arises if LY is constant as Y and r change, while Lr gets large as r gets small and Lr gets small as r gets large. LY remaining constant is consistent with Classical monetary theory, the idea being that people hold the fraction LY of any increase in income in the form of money. Since r represents the opportunity cost of holding money, it also makes sense that Lr decreases as r increases. At low interest rates, a one percent increase in the interest rate would represent a large percentage increase in the cost of holding money. Thus, a small increase in the interest rate would encourage people to significantly reduce their money holdings. Alternatively, a small decrease would greatly increase people’s willingness to hold money. This situation is known as a liquidity trap. At high interest rates, people are already economizing on their cash balances. Thus, an increase in the interest rate will not lead to much of a decrease in the willingness to hold money. In contrast to the liquidity trap, here the economy is liquidity constrained. Hicks summarized these conditions by saying (1) there is some minimum below which the interest rate is unlikely to go and (2) there is a maximum level of income that can be financed with a given money supply. These assumptions generate the LM curve shown in Figure 7.2. The multiplier Figure 7.2: The LM Curve r LM 96 Chapter 7: IS-LM and Aggregate Demand L Slope Y Lr Y r 1 r L 0 . The 0 and P Lr M PLr first multiplier indicates that an increase in the money supply reduces the interest rate level, while the second multiplier indicates that inflation increases the interest rate level. Because Y is being held constant as these changes occur, these multipliers also indicate that an increase in the money supply shifts the LM curve to the right---r increases with Y fixed, while an increase in the price level shifts the LM curve to the left---r decreases with Y fixed. Equation (7.18) also generates the multipliers Figure 7.3 illustrates the curve shift associated with a money supply increase. Notice that the money supply increase allows for a higher real income level. However, the minimum real interest rate level remains unchanged. This is a Keynesian notion which remains widely held today: An increase in the money supply can be helpful in that it can relax a liquidity constraint thereby allowing for an increase in real income, however the ability of an increase in the money supply to decrease interest rates is limited. Figure 7.3: Effect of a Money Supply Increase r LM1 LM 2 Y 97 Liquidity Trap Liquidity Constrained Chapter 7: IS-LM and Aggregate Demand 7.5 Analyzing the IS-LM Model Using Comparative Static Analysis 7.5.1 Equilibrium and the Implicit Function Theorem Hicks used his IS-LM model to explain why Keynes and Classical economists differed as to expected effect of an increase in the inducement to invest. The Classical perspective was that an increase in the inducement to invest would increase the interest rate level with no change in output, while Keynes claimed that an increase in the inducement to invest would lead to an increase in real income with no change in the interest rate level. To formally examine a change in the inducement to invest, let be an exogenous variable which measures the willingness to invest at the going interest rate, and expand the investment function (7.7) so that (7.19) I I (r , ) , Inducement to Invest Animal Spirits Ir 0 , I 0 . The variable can be thought of as a measure of Keynes’ animal spirits. Keynes argued that investment is volatile partly because the willingness to invest is partly driven by unexplainable, spontaneous “animal spirits.” Using the investment function (7.19), the IS-LM model can be formally presented as: (7.20) Y C (Y T ) I (r , ) G NX IS equation (7.21) PL(Y ,r ) M LM equation Variables (8): Y, T, r, , G, NX, P, M Endogenous (2): r, Y Exogenous (6): T, G, M, , NX, P The IS-LM diagram introduced by Hicks is obtained by plotting the IS and LM curves in the same space. Figure 7.4: The IS-LM Diagram r LM r1 IS 98 Chapter 7: IS-LM and Aggregate Demand Y1 Y Figure 7.4 illustrates how the two equations (7.20) and (7.21) determine the economy’s real interest rate level and real income level. The IS curve shows the interest rate-income combinations which satisfy the IS equation (7.20). The LM curve shows the interest rate-income combinations which satisfy the LM equation (7.21). Thus, only the interest rate income combination (Y1 ,r1) satisfies (7.20) and (7.21) simultaneously. It is this combination which the IS-LM model predicts for the economy. A more careful analysis first requires the use of the implicit function theorem. As it is usually stated, the implicit function theorem applies to a system of equations presented in the form: F 1 ( y1 ,..., yn ; x1 ,..., xm ) 0 F 2 ( y1 ,..., yn ; x1 ,..., xm ) 0 (7.22) n F ( y1 ,..., yn ; x1 ,..., xm ) 0 where the n equations implicitly define a set of n functions which determine each of the n endogenous variables (y1,...,yn) in terms of the m exogenous variables (x1,...,xn). Rewriting the IS and LM equations in this form gives the system A1 (Y , r ; G , T , M , , NX , P ) Y C (Y T ) I (r , ) G NX 0 (7.23) A2 (Y , r ; G , T , M , , NX , P ) PL (Y , r ) M 0 For the system (7.22), if some technical details are satisfied1, the implicit function theorem tells us that the endogenous y variables are determined implicitly as functions of the exogenous x variables if the Jacobian determinant is not equal to zero. For the general system (7.22) the Jacobian determinant is given by F 1 y1 F 2 (7.24) J y 1 F 1 F 1 y2 yn 2 F F 2 y2 yn F n y1 F n F n y2 yn 1 See Fundamental Methods of Mathematical Economics by Alpha Chiang for a careful, understandable presentation of the implicit function theorem. 99 Implicit Function Theorem Chapter 7: IS-LM and Aggregate Demand For our system (7.23), the Jacobian determinant is Jacobian Determinant Linear System Cramer’s Rule A1 (7.25) A Y2 A Y A1 r 1 C' I r PL [1 C'] PI L 0 . r r Y A2 PLY PLr r Because the determinant (7.25) for our system is negative, we know our system has a solution. 7.5.2 Comparative Static Multipliers and Cramer’s Rule When an exogenous variable changes, the equilibrium described by the IS and LM equations is perturbed. However, under the assumption that the IS and LM equations always hold, the economy must always be in equilibrium. Thus, the endogenous variables Y and r must adjust to new equilibrium levels. Comparative static analysis involves comparing equilibrium states. The analysis is static because time plays no essential role. That is, we do not consider the disequilibrium path the economy might follow as it moves from one equilibrium to another. This is unfortunate since many interesting things happen can happen out of equilibrium. However, ignoring disequilibrium greatly simplifies the analysis while still providing a forecast of change consistent with the assumption that the new equilibrium will be reached. To obtain comparative statics multipliers for our system, the next step is to totally differentiate the system (7.23). Doing so, one obtains Comparative Static Analysis Comparative Statics Multipliers (7.26) dY C' dY C' dT Ir dr I d dG dNX 0 PLY dY PLr dr LdP dM 0 The system (7.26) contains two equations and two endogenous differentials---dY and dr. It is a linear system because the differentials dY and dr only appear as coefficients. There are two ways in which this linear system can be solved. One method is by using substitution. Using substitution is easier for systems with fewer equations and fewer variables. However, it can become a tedious art for more complicated systems. Here, we demonstrate a second method called Cramer’s Rule---a general, systematic method for solving a linear system. To use Cramer’s rule, the linear system must first arranged in the matrix form Ax=d. The matrix A has n rows and n columns, corresponding to the system’s n equations and n endogenous variables. The vector x has n rows and consists of the system’s n endogenous variables. The vector d has n rows and consists of all terms which do not contain an endogenous variable. Immediately below, we show this matrix system in general for two equation system and then show how our system (7.26) is written in this matrix form: 100 Chapter 7: IS-LM and Aggregate Demand a11 a12 x1 d1 a 21 a22 x2 d2 (7.27) I r dY C ' dT I d dG dNX PLr dr LdP dM 1 C' PL Y Here, the system’s endogenous variables are the endogenous differentials; i.e., x1=dY and x2=dr. The second step in using Cramer’s rule is to calculate the determinant A . Notice that this determinant is the Jacobian (7.25). Thus, one nice thing about using Cramer’s rule is that the main requirement of the implicit function theorem can be checked along the way. From (7.25) above, we know that A [1 C'] PLr I r PLY 0 . The third step involves calculating the determinants A1 and A2 , where the matrix Ai is created by replacing column i in the matrix A by the vector d. For our system the determinants A1 and A2 are given by d1 d2 (7.28) A1 a12 C ' dT I d dG d [ NX ] I r a22 LdP dM PLr PLr C' dT PLr I d PLr dG PLr d[ NX ] Ir LdP Ir dM and (7.29) A2 a11 d1 1 C ' C ' dT I d dG d [ NX ] a12 d 2 PLY LdP dM [1 C']dM [1 C']LdP PLY C' dT PLY I d PLY dG PLY d[ NX ] The final step in using Cramer’s rule is to construct the solutions for the endogenous variables. In general, and for our example system, this is done as follows: (7.30) x1 A1 A dY PLr C ' PL I PL PL I L dT r d r dG r d [ NX ] r dP A A A A A Ir dM A 101 Chapter 7: IS-LM and Aggregate Demand and (7.31) x2 A2 A dr PLY C ' PL I PL PL dT Y d Y dG Y d [ NX ] A A A A [1 C'] L [1 C'] dP dM A A As before, the multipliers are the coefficients on the exogenous differentials. Because A 0 , the sign of the particular multiplier is the opposite sign of the coefficient’s numerator. What can be learned from these multipliers? This question is explored in the next few sections. 7.5.3 Changes in Fiscal Policy The term fiscal policy is used to describe government policy regarding government purchases and net taxes. To begin interpreting the multipliers, note that the multipliers for government purchases and net exports are the same. Presenting these multipliers in detail, we have (7.32) Y Y 1 0 and G [ NX ] [1 C '] I LY r Lr (7.33) LY Lr r r 0. LY G [ NX ] [1 C'] I r Lr Fiscal Policy An increase in G or NX shifts the IS curve up or to the right, which increases Y and r. The multipliers for G and NX are identical because the exogenous shock to the economy is precisely the same. A one dollar increase in G and a one dollar increase in NX each increase aggregate expenditure by one dollar. Such an increase in spending is sometimes referred to as “autonomous” because G and NX are exogenous variables, meaning any change in their levels is independent of changes in any other variables in the model. For this reason, the multiplier (7.32) is often called the autonomous spending multiplier. The increase in spending caused by an increase in G or NX is a direct increase. This direct increase can generate indirect increases in spending because an increase in one person’s spending becomes an increase in another’s income, which increases consumption spending, which further increases income. For any direct one dollar increase in spending, the autonomous spending multiplier gives the total direct and indirect 102 Chapter 7: IS-LM and Aggregate Demand increase in spending generated as direct increase ripples through the economy. Examining (7.32) notice that the size of this total increase depends upon the slope of the LM curve -LY/Lr . For a given marginal propensity to consume C’, the autonomous spending multiplier is as large as it can be---1/[1-C’]>1---when the slope of the LM curve is equal to zero. To the contrary, as the slope of the LM curve -LY / Lr increases, the autonomous spending multiplier eventually approaches zero. Examining (7.33) note that the extent to which the interest rate level increases also depends upon LY / Lr. The ratio -LY / Lr gives the increase in the interest rate which occurs when the real income level increases by one unit, assuming the monetary authority does not change the money supply. This interest rate increase “crowds out” investment spending equal to IrLY / Lr . This crowding out is minimal when LY / Lr is near zero. However, when LY/Lr is infinitely large the crowding out is complete---any increase in government purchases G or net exports NX is exactly completely offset by a decrease in investment I so that consumption spending C and total spending Y remain unchanged. Expansionary Policy Contractionary Policy Balanced Autonomous Budget Spending Multiplier Multiplier Next, consider an increase in net taxes T. A one dollar increase in net taxes reduces disposable income Y-T by one dollar, meaning consumption decreases by C’ dollars. This direct decrease in aggregate demand then ripples through the economy. Consequently, the tax multiplier for income is simply -C’ times the autonomous spending multiplier (7.32). In detail the tax multipliers are C' (7.34) LY Lr Y C' r 0 and 0. T [1 C '] I LY T [1 C'] I LY r r Lr Lr An change in fiscal policy affects the budget deficit G-T. An expansionary policy is one the generates an increase in aggregate demand. Alternatively, a contractionary policy decreases aggregate demand. Note that an increase in government purchases is expansionary, whereas an increase in net taxes is contractionary. What happens to total spending if government purchases and net taxes each increase by a dollar so that the budget deficit remains unchanged? We can consider this question by examining the so called balanced budget multiplier, (7.35) Y Y 1 C' 0 G T [1 C ' ] I LY r Lr The balanced budget multiplier is positive because a dollar increase in government purchases directly increases aggregate demand by one dollar, whereas a one dollar increase in net taxes generates a direct decrease in aggregate demand of C’ dollars as explained above. Thus, as long as there is a propensity to save---i.e., C’<1, a balanced 103 Crowding Out Chapter 7: IS-LM and Aggregate Demand budget policy is expansionary while G and T are increasing and contractionary while G and T are decreasing. 7.5.4 Change in the Inducement to Invest Next, consider an increase in the inducement to invest, something of particular interest to Hicks. The coefficients on d in (7.30) and (7.31) respectively tells us how a one unit increase in the inducement to invest affects the real income level and real interest rate level. Presenting these multipliers in detail, we have (7.32) Y I 0 [1 C '] I LY r and Lr LY I Lr r 0. L [1 C '] I r Y Lr An increase in the inducement to invest shifts the IS curve up or to the right, increasing Y and r. More precisely, a one unit increase in the inducement to invest leads to a direct increase in investment equal to I. The spending multiplier (7.31) then multiplies this direct investment spending increase into the total given by the income multiplier in (7.32). Again, the extent to which interest rates increase depends the slope of the LM curve LY/Lr. Recognizing this, Hicks argued that the main difference between Keynes’ “general theory” and Classical theory was a different assumption about the slope of the LM curve. As noted above, the extreme version of Classical monetary theory, which Hicks called “crude quantity theory,” contains only a transactions demand for money, and any effects which the interest rate level might have on money demand are ignored. This implies Lr = 0 and -LY/Lr = +. With the economy liquidity constrained, the “crude” Classical theory predicts that an increase in the inducement to invest will increase the interest rate level without much change in real income. To the contrary, Keynes predicted an increase in real income, and consequently an increase in employment, with little change in the interest rate level. Hicks showed that Keynes’ prediction is consistent with an economy in a liquidity trap, where Lr is near - and -LY/Lr is near zero---an assumption crude to the same degree as that made in the Classical’s crude quantity theory but at the opposite extreme. Thus, the brilliance of Hicks’ IS-LM invention was that it offered a more general theory than Keynes’ general theory---one that includes the Keynesian and Classical extremes as special cases. 7.5.5 Changes in Monetary Policy To examine the impact of money, consider the model’s money multipliers 104 Chapter 7: IS-LM and Aggregate Demand (7.33) Ir PLr Y 0 M [1 C'] I LY r and Lr 1 C' PLr r 0. M [1 C'] I LY r Lr Keynes stressed that an increase in the money supply will have little positive impact on a depressed economy because people will tend to hold the additional money rather than lend it or spend it due to the high levels of risk and uncertainty. As mentioned, this liquidity trap idea can be modeled by making Lr large. As Lr increases, both of the money multipliers in (7.33) approach zero. That is, as Keynes suggested, expansionary monetary policy is useless in a liquidity trap because it affects neither the income level nor the interest rate level. Because a liquidity trap would likely occur only in a severe recession, Hicks referred to Keynes general theory as the “Economics of Depression.” Hicks also used the IS-LM apparatus to show that a money supply increase can be useful if the economy is operating in the liquidity constrained condition where the LM curve is near vertical. Adopting the Classical extreme Lr = 0, dY/dM becomes 1/PLY >0. Notice that there is no interest rate parameter in this multiplier. Here, real income increases directly in response to the money supply increase rather than indirectly through an increase in investment caused by lower interest rates. In contrast to his characterization of “crude” Keynesian economics as the Economics of Depression, this “crude” Classical view could be called the “Economics of Prosperity,” a state where people have so economized on their cash balances that the opportunity cost of holding money is ignored. Demand-Side Supply-Side 7.5.6 IS-LM and Aggregate Demand An increase in the price level P shifts the LM curve up or to the left, generating a decrease in Y and an increase in r. In detail, the price multiplier for income is (7.34) Ir L PLr Y 0 L P [1 C '] I r Y Lr The increase in the price level decreases the real money supply creating an excess demand for money. The decrease in the real interest rate necessary to eliminate this initial excess money demand is L/PLr. This generates a direct decrease in investment spending equal to IrL/PLr . The income multiplier (7.34) captures the total increase in spending which results as the direct decrease in investment spending ripples through the economy. Note that all of the multipliers in the IS-LM model are obtained without any reference to the production process. This is because the IS-LM model only represents the demand- 105 Aggregate Demand Curve Chapter 7: IS-LM and Aggregate Demand side of the economy. The demand-side of the economy determines aggregate demand by recognizing the constraints on the willingness and ability to buy. Alternatively, the supply side of the economy determines aggregate supply by recognizing the constraints on the willingness and ability to produce and sell. Because it does not recognize the real constraints on the ability to produce in the form of limited labor, capital, raw materials, and technology, the IS-LM model does not contain a supply-side. The equilibrium Y value is the aggregate demand level determined by the IS-LM model. The willingness and ability to buy is constrained by both current income and accumulated assets. The product market, represented by the IS equation, captures the effect of current income on the willingness and ability to buy. The money market, represented by the LM equation, captures the effect of assets. If either the product market or money market is in disequilibrium, the buying power present in the economy changes as the real income and real asset levels change. Only when both markets are in equilibrium is total willingness/ability to buy---i.e., aggregate demand---in a state of equilibrium. The aggregate demand curve is the set of (P,Y) combinations associated with a simultaneous product market-money market equilibrium. The multiplier (7.34) tells us that the slope of the aggregate demand curve is negative as shown in Figure 7.5. By presenting Y on the vertical axis and P on the horizontal axis, we go against economic tradition. Nearly all economics texts do the opposite. However, we are considering Y as being dependent upon P, meaning Figure 7.5 is mathematically proper. As we have drawn it, the slope of the aggregate demand curve becomes steeper---i.e., output is more sensitive to a change in the price level---when the autonomous spending multiplier is larger, when investment is more sensitive to a change in the interest rate level, and when money demand is less sensitive to a change in the interest rate level. Figure 7.5: The Aggregate Demand Curve Y AD P The IS-LM model provides an answer to the question posed at the beginning of this chapter. Where does aggregate demand come from? The IS-LM model tells us that it depends upon 106 Chapter 7: IS-LM and Aggregate Demand the (exogenous) variables T,G, M, , NX, P; the particular form of the consumption function as characterized by the marginal propensity to consume C’; the particular form of the investment function as characterized by the sensitivity of investment to interest rates Ir and the sensitivity of investment to animal spirits I; the particular form of the money function as characterized by the sensitivity of money demand to interest rates Lr and the sensitivity of money demand to income LY. To construct a more compete macro model, we must construct a model of the supply side and then combine it with this demand-side model. A supply-side model is developed in the next chapter. Key Words: Chapter 7 Aggregate Demand LM-Curve Consumption Function Savings Function Marginal Efficiency of Capital IS Equation Capital Market Exogenous Variable Multipliers Transactions motive Speculative motive LM Equation Liquidity Constrained Animal Spirits Jacobian determinant Comparative Statics Multipliers Cramer’s Rule Crowding out Contractionary Policy Demand-Side Aggregate Demand Curve Key Concepts: Chapter 7 Review Questions: Chapter 7 107 IS-Curve Aggregate Supply Marginal Propensity to Consume Marginal Propensity to Save Investment Function Product Market Endogenous Variable Implicit Function Money Demand Precautionary motive Money market Liquidity Trap Inducement to Invest Implicit Function Theorem Comparative Static Analysis Linear System Autonomous spending multiplier Expansionary Policy Balanced Budget Multiplier Supply-Side