PLEASE NOTE: These basic lecture notes on HT Chapter 6 are required for Econ 302. If possible, students should read these notes prior to attending class lectures on HT chapter 6. A more advanced detailed version of these notes is also available on this web site; the advanced notes are recommended but not required.
The HT story regarding the movement of the economy over time relies on the following key assumptions, based on empirical observation:
FE Line P . | ad' ad . | ad' ad . | ad' ad . | ad' ad . | ad . P(T) |- - - - -ad'- - - - E- - - - - - - - - - - - | ad' . ad | ad' . ad P(T+1)|- - - - - - - - - - E'- - - - -ad- - - - - - | . ad' ----------------------------------------------- Y 0 Y(T) Y*(T) Figure 6.1: Slow Price Adjustment to a Negative Demand Shock.
The sequence of events depicted in Figure 6.1 is as follows. At the beginning of some period T, the economy is in equilibrium at point E with actual GDP equal to the potential GDP level Y*(T). A negative demand shock shifts the aggregate demand curve to the left, from AD to AD' (lower Y for each P). The economy's output decreases from Y*(T) to Y(T), resulting in the opening up of a negative GDP gap, [Y(T)-Y*T(T)]/Y*(T), indicating sluggish demand relative to potential production. However, producers do not immediately change their prices. In the subsequent period T+1, the price level P(T) begins to fall as producers lower their prices in response to the persistently negative GDP gap. This results in a downward movement along the new AD' curve. The downward movement along AD' continues until the GDP gap is eliminated (Y=Y*); this occurs at point E'.
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 AD curve: namely, the "spending balance relation" describing a situation of product market equilibrium in which the aggregate demand for newly produced final goods and services equals the actual supply of newly produced final goods and services. As will be clarified below, this spending balance relation is equivalent to the condition that actual saving equals planned saving equals planned investment equals actual investment. For this reason, the spending balance relation is ofen simply referred to as the "IS relation."
Specifically, in Chapter 6, HT concentrate on two principal behavioral components underlying the spending balance relation: planned consumption as a function of income; and planned net exports as a function of income. As in HT6, the dependence of variables on the time period T is omitted below for expositional simplicity.
Recall the form of the National Income Accounting Identity. For any specified time period T,
Y = C + I + G + NE , where Y = realized GDP; C = realized consumption; I = realized investment; G = realized government expenditures on newly produced final goods and services; NE = realized 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 that 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, and ROW.
Following HT Chapter 6, the notes below focus specifically on how planned consumption and planned net exports depend on income. Subsequent chapters will focus on the remaining needed behavioral relations.
Determinants of Planned Consumption:
Given any period T, the consumption function for period T describes the total period T consumption demand of all households (families) in the economy as a function of one or more key explanatory variables.
In his pathbreaking work, the 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 the income at the disposal of households after subtraction of personal taxes. This postulate has been borne out by extensive empirical research. The simple version of this postulate used by HT is as follows:
(1) [consumption function] C^D = a + b[1-t]Y , where [1-t]Y = disposable income, i.e., income Y minus income taxes tY, where the tax rate t is strictly positive and strictly less than 1; C^D = consumption demand, i.e., planned spending of households on newly produced final goods and services; a = nonnegative coefficient denoting the level of consumption demand when disposable income is zero; b = coefficient denoting the "marginal propensity to consume" out of 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 C^D c | c | c C^D = a + b[1-t]Y | c | c with slope dC^D/dY = b[1-t] | c | c | c a | Intercept = a | | -------------------------------------------------------- Y 0 Fig. 6.2: An Illustrative Consumption Function
Note that the consumption function shifts in response to a change in the intercept a. For example, if a increases, then the consumption function shifts up (higher C^D for each Y). Also, the consumption function rotates around a in response to any change in either the marginal propensity to consume, b, or the income tax rate, t; for a change in either b or t affects the slope of the consumption function without affecting the intercept a. For example, an increase in b rotates up the consumption function (larger C^D for each positive Y); for the slope of the consumption function is now a larger positive number but the intercept a is unchanged.
Spending Balance for the Simple Case of Exogenously Given Investment, Government Expenditure, and Net Exports:
We will now construct a simple model of an economy whose market for final goods and services is in equilibrium, in the sense that the aggregate demand for newly produced final goods and services equals the actual supply.
Suppose for simplicity that the values I, G, and NE for investment, government expenditures, and net exports take on given positive values at the beginning of period T. The aggregate demand Y^D for newly produced final goods and services then takes the form
(2) [aggregate demand] Y^D = C^D + I + G + NEAlso, recall the national income accounting identity for the actual supply Y of newly produced final goods and services:
(3) [actual supply] Y = C + I + G + NE,where C denotes realized consumption. Finally, the following relation asserts that the aggregate demand Y^D for newly produced final goods and services equals the actual supply Y:
(4) [product market equilibrium] Y^D = Y
Using equation (4), one can replace Y^D in equation (2) by Y. Equations (2) and (3) together then imply that C^D = C. Consequently, equations (1) through (4) characterizing equilibrium in the product market reduce down to just two distinct equations in two unknowns, Y and C. HT refer to these two equations as the "spending balance equations."
Spending Balance Equations [Compare HT (6-1) and (6-3)]:
(M1) Y = C + I + G + NE; (M2) C = a + b[1-t]Y .
Classification of Variables:
KEY DEFINITION: The economy will be said to be in spending balance if it satisfies equations (M1) and (M2). The economic interpretation of spending balance is that the product market of the economy is in equilibrium in the sense that aggregate demand equals actual supply.
Obtaining Spending Balance Solutions for Y and C:
The model described by equations (M1) and (M2) represents two equations in the two unknown endogenous variables C and Y. The model-determined solutions for Y and C, denoted by Y^o and C^o, respectively, can be found by solving equations (M1) and (M2) using the following steps.
First, substitute (M2) into (M1) to obtain one equation in the one unknown Y:
(5) Y = (a + b[1-t]Y) + I + G + NE.
Collecting terms in Y, and solving for Y, yields the spending balance solution for Y [compare HT equation (6-5)]:
Spending balance solution for income Y:
a + I + G + NE (6) Y^o = --------------------- 1 - b[1-t]
Note that the denominator of the right side ratio in (6) is strictly positive and strictly less than 1; for, by assumption, b and t are both strictly positive and strictly less than 1. Substituting Y^o for Y in (M2) then yields the spending balance solution for C:
Spending balance solution for consumption C:
(7) C^o = a + b[1-t]Y^o .
We need to step back for a moment and examine 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 adjust their production to changing demand conditions at the prevailing set of prices.
Note, in particular, that each term on the right side of (6) is an exogenous variable coming from the demand side of the economy: namely, one has the exogenously given coefficients (a,b,t) appearing in the consumption function (1), and the exogenously given investment level I, government expenditure level G, and net export level NE appearing in the aggregate demand relation (2). On the other hand, the left side of (6) is the solution value Y^o for actual supply. A change in any of the exogenous variables on the right side of equation (6) results in a corresponding change in Y^o on the left side of (6). Thus, in any given period T (i.e., in any "short run"), supply adjusts to demand. This is the essence of the HT story.
Given the explicit spending balance solution (6) for Y, one can now carry out comparative static experiments to determine how this solution varies in response to a change in any one of the variables on the right side of (6).
Using several examples, it will be shown how one can think of a change in a right hand exogenous variable in (6) as leading to a round of changes in the left hand solution Y^o, as if a dynamic process were being modelled. Note that HT do not actually present an explicit dynamic model of the process by which a new spending balance is achieved following a change in an exogenous variable. Rather, they simply assume it is achieved in the same time period that the disturbance takes place.
Example A: The Investment Multiplier
Suppose investment I decreases. Differentiating the Y^o solution in (6) with respect to a change in I, one obtains the investment multiplier:
dY^o 1 (8) ------ = ------------- , dI 1 - b[1-t]where the term 1/(1-b[1-t]) in (8), called the investment multiplier, is strictly greater than 1.
The differential relation (8) can also be expressed in "finite difference" form as
1 (9) Delta(Y) = ------------ x Delta(I) , 1 - b[1-t]where Delta(Y) = [Y' - Y] denotes the incremental change in actual supply from Y to Y' that results from an incremental change Delta(I) = [I'-I] in investment from I to I'.
Since (1 - b[1-t]) is strictly positive and strictly less than 1, the investment multiplier (8) 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 that is even larger in magnitude.
Note the importance of the latter observation. Relatively small changes in investment can lead to relatively large changes in 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 (M1) and (M2) used to determine the spending balance solutions for Y and C:
Y = C + I + G + NE ; C = a + b[1-t]Y .
The initial impact of a decrease in investment I is to decrease Y on a dollar for dollar basis with no change in C or G or NE. Suppose, for example, that I decreases by $100 (in base year dollars).
Y = C + I + G + NE down by down by $100 $100The corresponding reduction in Y then 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] x $100 by $100 dollars dollarsThe decrease in consumption further reduces Y by b[1-t]$100. Thus, the initial dollar decrease in I leads to an even further 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.
By assumption, however, 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-qwhere 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 subsequent decreases in Y is given by
- $100 - $100 --------- = ------------ , 1-q 1 - b[1-t]which is simply the amount predicted using relation (9) with Delta(I) = -$100.
Example B: The Government Multiplier
Define the government multiplier to be the change in Y corresponding to any change in G, given fixed levels of I and NE. The government multiplier can be found by differentiating Y^o in (6) with respect to G:
dY^o 1 (10) ------- = ------------- , dG 1 - b[1-t]where the right-hand term in (10) is strictly greater than 1. Note that the government multiplier (10) coincides with the investment multiplier (8) for this simple model of an economy.
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 (6) with respect to t, one finds that
dY^o - bY^o (11) ---- = -------------- < 0 . dt (1 - b[1-t])
Now consider the case of an economy for which net export demand depends on income. Let
NE^D = [EX^D - IM^D] = net export demand.
As a matter of empirical observation, NE^D generally decreases in the short run with increases in income Y, all other things remaining equal, because IM^D increases with increases in Y but EX^D does not significantly change in response to changes in Y.
Suppose for simplicity that import demand is a linear function of income:
IM^D = mY ,where m is a positive constant referred to as the marginal propensity to import. Suppose, also, that export demand for period T takes on some exogenously given positive constant value g:
EX^D = g .Then net export demand for time T take the form
(12) NE^D = g - mY .
Equation (12) implies that net export demand declines when income increases, in accordance with empirical observation. [Dependence of net export demand on the exchange rate is taken up in HT Chapter 12.]
Assuming, as before, that I and G are exogenously given, the model for the determination of spending balance with income-dependent net exports can now be written out as follows:
(m1) [aggregate demand] Y^D = C^D + I + G + NE^D ; (m2) [consumption function] C^D = a + b[1-t]Y ; (m3) [net export demand] NE^D = g - mY ; (m4) [actual supply] Y = C + I + G + NE ; (m5) [AD = actual supply] Y^D = Y ; (m6) [demand=supply for NE] NE^D = NE .
Classification of Variables:
We can now proceed to reduce these six equations down, by substitution, until we obtain a precise expression for the spending balance level of Y that solves this model. Using the last three equations to substitute out for 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 .Then, substituting out for C and NE in (m1)+, using (m2)+ and (m3)+, one obtains one equation in the one unknown Y:
Y = (a + b[1-t]Y) + I + G + (g - mY) .Collecting terms in Y and solving for Y, one obtains the following spending balance solution for Y: [compare HT, equation (6-9)]:
Spending balance solution for Y 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] + mCompare (13) with the investment multiplier (8) and government multiplier (10) obtained for the original model. If m were equal to zero, these multipliers would all be the same. However, since m is assumed to be strictly positive, the multiplier (13) derived with income-dependent net exports is smaller than the multipliers (8) and (10) derived with exogenous net exports.
What is the intuitive reason for this difference?
Suppose I decreases by $100.00 (in base year dollars).
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.