Basic References: Hall and Taylor, Chapter 6; Study Guide, Chapter 6. #SHORT-RUN FLUCTUATIONS AND SPENDING BALANCE# OVERVIEW OF THE HT SHORT-RUN FLUCTUATION MODEL #The HT Labor and Capital Hoarding Story#: a) Under normal business conditions, most firms operate with some excess capital capacity (84% capacity level on average) and also some excess labor capacity. b) In response to unanticipated changes in demand for their goods and services, the first response of firms is to adjust their production levels to satisfy this changed demand. c) These initial adjustments to production in response to unanticipated changes in demand are accomplished by means of #temporary# adjustments to labor and capital inputs. d) In particular, attempts might be made to change the effort levels (output per hour) of current employees, or the amount of overtime required of current employees, or the amount of temporary help, or the number of employees on temporary layoff, rather than immediately engaging in the hiring or firing of regular employees; and attempts might be made to change the #intensity# with which current capital inputs are #used# rather than immediately engaging in the purchasing or selling of capital equipment. e) Only after a change in demand persists for some length of time, so that firms are convinced the change is permanent rather than temporary, will firms act to adjust the amounts of their labor and capital inputs on a more permanent basis--- through the firing or hiring of regular employees and the purchase or sale of capital equipment. f) Prices are "sticky" compared to production levels. That is, the adjustment of prices in response to changes in demand or other aspects of the economic environment occurs gradually whereas the adjustment of production levels occurs much more quickly. g) A macro implication of this slow price adjustment is that changes in aggregate demand first result in changes in aggregate output Y and only later result in changes in the general price level P. FE Line P . | . | . | . | . | B . E P(T) |........................................... | . | . AD P(T+1)|........................................... | . AD' ----------------------------------------------- Y 0 Y(T) Y(T+1) Y*(T) Figure 6.1: Slow Price Adjustment to a Negative Demand Shock. The economy is originally in equilibrium at E. A negative demand shock shifts the aggregate demand curve from AD to AD', leading to excess supply (excess capacity to produce) at B. The price level P(T) begins to fall in response to this excess supply in the subsequent period T+1, leading to a downward movement along the new AD' curve. In Chapter 6, HT begin the construction of this short-run fluctuation model by focusing on a key concept entering into the construction of the aggregate demand curve; namely, the IS (saving=investment) or spending balance relation. The spending balance relation describes the condition that demand equal supply in the product market---i.e., the market for all newly produced final goods and services---or equivalently, the condition that #planned# savings be equal to #planned# investment. Specifically, in Chapter 6, HT concentrate on two principal behavioral components underlying the spending balance relation: planned consumption and planned net exports as functions of income. A more detailed treatment of these and other components entering into the spending balance relation is taken up in HT Chapter 7. For expositional simplicity, following HT, the dependence of variables on the time period T will be omitted throughout the following notes; that is, Y(T) will simply be written as Y, and similarly for all other time-dated variables. CONSTRUCTING THE IS (INCOME-SAVING) RELATION: Recall the form of the National Income Accounting Identity for any specified time period T: Y = C + I + G + NE , where Y = realized period T real GDP; C = realized period-T real consumption; I = realized period-T real investment; G = realized period-T real government expenditures on newly produced final goods and services; NE = realized period-T real net exports . We now want to consider descriptions of how different types of economic agents in the economy #plan# their purchases of goods and services. That is, we want to develop the planning concepts which correspond to each spending category appearing in the national income accounting identity. To do this, we need to make explicit assumptions concerning the behavior of consumers, firms, government, financial intermediaries, and ROW. Following HT Chapter 6, the notes below focus specifically on how planned consumption and planned net exports depend on income. Determinants of Planned Consumption Given any period T, the #consumption function# for period T describes the total consumption demand of all households (i.e., families) in the economy as a function of one or more key explanatory variables. In his pathbreaking work #General Theory of Employment, Interest, and Money# (1936), John Maynard Keynes argued that, in the aggregate, the primary variable determining the consumption demand of households is expected disposable income, i.e., the income that households expect to receive #after# subtraction of personal taxes. This postulate has been borne out by extensive empirical research. For expositional simplicity, following HT, it will be assumed that V (net factor income and transfer payments from abroad), F (government transfers to the private sector), and N (interest payments on the public debt), as well as various other miscellaneous items (see HT Tables 2-5 and 2-6), are zero for the economy at hand, so that the real GDP Y also measures the total real personal income received by households before taxes. Also, it will be assumed that the only tax faced by households is an income tax assessed at some flat rate t---for example, t = 0.17. Finally, it will be assumed that the consumption function is a simple linear function of disposable income. Specifically, in line with these assumptions, it will be assumed that the consumption function describing consumption demand as a function of expected disposable income takes the form C^D = a + bY`d^e = a + b[1-t]Y^e , where Y`d = [1-t]Y = real disposable income for period T, i.e., real income Y minus income tax payments tY, where t is strictly positive and strictly less than 1; Y`d^e = [1-t]Y^e = expected real disposable income for period T; C^D = period-T consumption demand, i.e., planned period-T spending of households on newly produced final goods and services; a = nonnegative coefficient denoting the level of consumption demand when real disposable income is zero; b = coefficient denoting the #marginal propensity to consume# out of real disposable income, where b is strictly positive and strictly less than 1. The restrictions on the marginal propensity to consume, b, can be motivated as follows. As a rule of thumb [and in keeping with Keynes' consumption postulates], one would expect on average to see households spending only a #fraction# of every dollar of their disposable income on consumption spending, the rest being allocated to saving. For this reason one would expect the coefficient b to lie between zero and one. C^D | | C^D = a + b[1-t]Y^e, | with slope: dC^D/dY^e = b[1-t] | | | | a | | intercept = a = [value of C^D when Y^e = 0] | -------------------------------------------------------- Y^e 0 Fig. 6.2: The Consumption Function in the C^D-Y^e plane Note that the consumption function #shifts# up or down in response to an increase or decrease in the intercept coefficient a, and #rotates# in response to any change in the slope term b[1-t], that is, in response to any change in the marginal propensity to consume, b, or the income tax rate, t. #Spending Balance for the Simple Case of Exogenously Given Investment, Government Expenditure, and Net Exports#: Suppose, for initial analytic simplicity, that the values I, G, and NE for real investment, real government expenditures, and real net exports take on given positive values at the beginning of period T, so that only consumption and income remain to be determined. Define #real aggregate demand# [aggregate planned real spending] for period T, denoted by Y^D, as follows: (1) [#real aggregate demand#] Y^D = C^D + I + G + NE , where (2) [#real consumption demand#] C^D = a + b[1-t]Y^e . Recall, also, the accounting identity for realized real GDP, here assumed to coincide with realized real income: (3) [#realized real GDP (income)#] Y = C + I + G + NE, where C denotes realized real consumption. The economy is said to have attained a #spending balance# for period T if, in addition to (1)-(3), the following two equilibrium conditions hold: (4) [#fulfilled expectations#] Y^e = Y [i.e., expected real income equals realized real income] (5) [#fulfilled plans#] Y^D = Y [i.e., real aggregate demand for goods and services equals the realized real supply of goods and services.] #Classification of Variables for Model (1)-(5)#: Exogenous Variables: [i.e., Known var's determined outside of model (1)-(5)] I, G, NE, a, b, t, with all terms strictly positive and b and t also strictly less than 1 Endogenous Variables: [i.e., Unknown var's to be determined by solving model (1)-(5)] Y^D, C^D, C, Y, Y^e (NOTE: No. of equations = No. of unknowns) Using equations (4) and (5), one can replace all occurrences of Y^D and Y^e in equations (1) through (3) by Y, which results in the following modified versions of equations (1) through (3): (1)* Y = C^D + I + G + NE (2)* C^D = a + b[1-t]Y (3)* Y = C + I + G + NE Finally, note that equations (1)* and (3)* together imply that C^D = C. Hence, substituting C for C^D in (1)*, one is left with the following two equations in the two unknowns Y and C: Basic Spending Balance Equations [Compare HT (6-1) and (6-3)]: (1)** Y = C + I + G + NE; (2)** C = a + b[1-t]Y . #Classification of Variables for Model (1)**-(2)**#: Exogenous Variables [Var's determined outside of model (1)**-(2)**]: I, G, NE, a, b, t, with all terms strictly positive and b and t also strictly less than 1 Endogenous Variables [Var's to be determined by model (1)**-(2)**]: C, Y [Note: Number of equations = Number of unknowns] #Algebraic Solution for the Basic Spending Balance Equations#: Model (1)**-(2)** represents two equations in two unknowns: namely, the two endogenous variables C and Y. The #spending balance solutions for Y and C#, denoted by Y^o and C^o, respectively, can be found by solving equations (1)** and (2)**. First, substitute (2)** into (1)** to obtain one equation in the one unknown Y: (6) Y = (a + b[1-t]Y) + I + G + NE. Manipulating terms in this expression, one obtains the following solution Y^o for real income [compare HT, equation (6-5)]: Spending balance solution for real income: a + I + G + NE (7) Y^o = --------------------- 1 - b[1-t] Note that the denominator of the right-side ratio in (7) is strictly positive and strictly less than 1 because, by assumption, b and t are both strictly positive and strictly less than 1. Substituting this Y^o solution into (2)**, one obtains the following solution C^o for real consumption: Spending balance solution for real consumption: (8) C^o = a + b[1-t]Y^o . We need to step back for a moment and ask what this all means. The spending balance solution for Y reflects the HT position that the demand side of the economy determines actual production "in the short run." The basic idea is that, in the short run (and within limits), producers can "instantaneously" adjust their production to changing demand conditions #at the prevailing set of prices#. Note, in particular, that each term on the right-side of (7) is an exogenously given term coming from the demand side of the economy---i.e., either an exogenously given coefficient appearing in the consumption demand function (a, b, or t), or the exogenously given demand for goods and services by firms (I) or government (G). In contrast, the left-side of (7) represents the supply-side of the economy---i.e., the realized level Y^o of real income (production). Any change in a right-hand term results in a corresponding change in Y^o. This can be predicted from the general form of the model equations (1)-(5); but the specific algebraic solution for Y^o allows one to determine the precise magnitude and sign of the resulting changes in Y^o. THE MULTIPLIER Given the explicit spending balance solution (7) for Y, one can now carry out #comparative static experiments# to determine how this solution varies with changes in any of the right-hand terms. Using several examples, it will be shown how one can think of changes in right-hand terms as leading to a round of changes in the left-hand real income solution Y^o, as if a dynamic process were being modelled. However, HT do not present a formal dynamic model of the process by which a new spending balance is achieved. Rather, they assume it is achieved in the same period of time during which the disturbance takes place. #Example A: The Investment Multiplier# Suppose investment I decreases. Differentiating the Y^o solution in (7) with respect to a change in I, one obtains the #investment multiplier#: dY^o 1 (9) ------ = ------------- , dI 1 - b[1-t] where the term 1/(1-b[1-t]) in (9), called the #investment multiplier#, is strictly greater than 1. The differential relation (9) can also be expressed in "finite difference" form as 1 (10) Delta(Y) = ------------ x Delta(I) , 1 - b[1-t] where Delta(Y) = [Y' - Y] denotes the incremental change in real income from Y to Y' that results from an incremental change Delta(I) = [I'-I] in real investment from I to I'. Since (1 - b[1-t]) is strictly positive and strictly less than 1, the investment multiplier (9) is positive and indeed is greater than 1. The fact that the multiplier is positive means that #increases# in I result in #increases# in Y, and #decreases# in I result in #decreases# in Y. The fact that the multiplier is greater than 1 means that any change in I translates into a change in Y^o which is even larger in magnitude. Note the importance of the latter observation. Relatively small changes in real investment spending can lead to relatively large changes in real income. This reflects the idea of Keynes that fluctuations in investment are an essential source of business cycle fluctuations. He thought that "animal spirits" guided a significant portion of investment, e.g., sudden correlated shifts in peoples' outlooks on future profitability of investment leading to bullish or bearish investment behavior. A more intuitive derivation for the investment multiplier will now be given. Consider again the two basic equations (1)** and (2)** used to determine the spending balance solution for Y and C: Y = C + I + G + NE ; C = a + b[1-t]Y . The initial impact of a decrease in investment spending is to decrease total spending Y on a dollar for dollar basis, for given levels of C and G. Suppose, for example, that I decreases by $100. Y = C + I + G + NE down by down by $100 $100 Next, however, the reduction in Y results in a decrease in the disposable income [1-t]Y of households, which in turn leads to a decrease in consumption. [The larger the marginal propensity to consume b, the larger the decrease in consumption.] C = a + b[1-t]Y down by down b[1-t]$100 by $100 dollars dollars The decrease in consumption #further# reduces total spending Y by b[1-t]$100. Thus, the initial dollar decrease in I leads to an even #further# dollar decrease in Y. And this is not the end of the story; for the secondary decrease in Y feeds back again into the consumption function to cause an even further decline in consumption, and so forth and so on. Note that, in each round, the additional decrease in Y is b[1-t] times the previous decrease in Y. Consequently, the total decrease in Y is the sum of the infinite sequence of effects -$100, b[1-t] x -$100, (b[1-t])^2 x -$100, .... etc. But, by assumption, the factor b[1-t] is strictly less than one in absolute value. By a basic theorem in mathematics, for any real number q that is strictly less than 1 in absolute value, one has 1 q^0 + q^1 + q^2 + ... = ----- , 1-q where q^0 = 1 by convention. (See HT, Footnote 4, page 166.) Consequently, letting q = b[1-t], and supposing I decreases by $100, the accumulated effect of all of the decreases in Y is given by - $100 - $100 --------- = ------------ , 1-q 1 - b[1-t] which is simply the amount predicted using relation (10) with Delta(I) = -$100. #Example B: The Government Spending Multiplier# Define the #government multiplier# to be the change in Y corresponding to any change in G, for fixed I and NE. The #government multiplier# can be found by differentiating Y^o in (7) with respect to G: dY^o 1 (11) ------- = ------------- , dG 1 - b[1-t] where the right-hand term in (11) is strictly greater than 1. Note for this simple linear model of an economy that the investment multiplier (9) coincides with the government multiplier (11). #Example C: The Tax Multiplier# Finally, what is the effect on Y^o of a change in the tax rate t? Differentiating Y^o in (7) with respect to t, one finds that dY^o - bY^o ---- = -------------- < 0 . dt (1 - b[1-t]) SPENDING BALANCE WHEN NET EXPORTS DEPEND ON INCOME Now consider the case of an economy for which net export demand depends on expected income. Let NE^D = [EX^D - IM^D] = (real) net export demand. As a matter of empirical observation, NE^D generally #decreases# in the short run with increases in expected real income Y^e, all other things remaining equal, because IM^D increases with increases in Y^e but EX^D does not significantly change in response to changes in Y^e. Suppose for simplicity that real import demand is a linear function of expected real income: IM^D = mY^e , where m is a positive constant referred to as the #marginal propensity to import#. Suppose, also, that real export demand for period T takes on some exogenously given positive constant value g, i.e., EX^D = g . Then real net export demand for time T take the form (12) NE^D = g - mY^e . Equation (12) implies that real net export demand #declines# when expected real income #increases#, in accordance with empirical observation. [Dependence of net export demand on the exchange rate will be taken up in HT Chapter 12.] Assuming, as before, that I and G are exogenously given, the model for the determination of spending balance can now be written out as follows: (m1) [aggregate demand] Y^D = C^D + I + G + NE^D ; (m2) [consumption demand] C^D = a + b[1-t]Y^e ; (m3) [net export demand] NE^D = g - mY^e ; (m4) [realized real GDP (income)] Y = C + I + G + NE ; (m5) [fulfilled expectations] Y^e = Y ; (m6) [fulfilled plans] Y^D = Y ; (m7) [fulfilled plans] NE^D = NE . #Classification of Variables for Model (m1)-(m7)#: Exogenous Variables [Var's determined outside of model (m1)-(m7)]: I,G,a,g,m,b,t, with all terms strictly positive and with b and t also strictly less than 1. Endogenous Variables [Var's determined by model (m1)-(m7)]: Y^D, C^D, NE^D, Y^e, Y, C, NE . We can now proceed to reduce these seven equations down, by substitution, until we obtain a precise expression for the spending balance income Y which solves this model. Using the last four equations to substitute out for Y^e, Y^D, C^D, and NE^D, one obtains (m1)+ Y = C + I + G + NE ; (m2)+ C = a + b[1-t]Y ; (m3)+ NE = g - mY . Thus, substituting out for C and NE in (m1)+, using (m2)+ and (m3)+, one obtains one equation in one unknown---the spending balance level of real income Y: Y = (a + b[1-t]Y) + I + G + (g - mY) . Solving this equation for Y, one obtains an explicit expression for the spending balance solution for Y as follows [compare HT, equation (6-9)]: Spending balance solution for real income in the modified model: a + I + G + g Y^+ = --------------------- . 1 - b[1-t] + m The investment and government multipliers associated with changes in I and G are therefore given by dY^+ dY^+ 1 (13) ----- = ------ = ----------------- . dI dG 1 - b[1-t] + m Compare (13) with the investment and government multipliers (9) and (11) obtained for the original model. If m were equal to zero, the multipliers would be the #same#. However, since m is assumed to be strictly positive, the multipliers (13) derived with income-dependent net exports are #smaller# than the multipliers (9) and (11) derived with exogenous net exports. WHAT IS THE INTUITIVE REASON FOR THIS DIFFERENCE? Suppose I decreases by $100.00. first round Y = C + I + G + NE down down by $100 by $100 second-round C = a + b[1-t]Y down by down by b[1-t]$100 $100 dollars NE = g - mY #up# by down by m$100 dollars $100 dollars Note that the #net# decrease in Y due to second-round effects is #smaller# than it was for the original model with exogenous net exports, since the decrease in Y stemming from the decrease in C is now partially offset by the gain in NE, hence in Y, due to the decrease in imports. This smaller change in Y leads to smaller changes in all subsequent rounds. The total effect on Y of the decrease in I by $100.00 is thus smaller for income-dependent net exports than it is for exogenous net exports. Notice, however, that the change from exogenous to income-dependent net exports does not change the #sign# of the investment and government multipliers. They are both still positive.