Short-Run Fluctuations and Spending Balance

**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.

- Basic References:
- HT Chapter 6
- Study Guide, Chapter 6

The HT story regarding the movement of the economy over time relies on the following key assumptions, based on empirical observation:

- 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 shocks (unanticipated changes) in the
demand for their goods and services, the first response of firms is
to adjust their production levels to satisfy their changed demand.
- c) These initial attempts to keep production in line with
changed demand are accomplished by means of
*temporary*adjustments to labor and capital usage. - 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 permanent hiring or firing of regular
employees. Similarly, attempts might be made to change the
intensity with which current capital inputs are used rather than
undertaking purchases or sales 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 change their output prices
and undertake more permanent adjustments to their labor and capital
inputs.
- f) Consequently, output prices are "sticky" compared to production
levels. That is, the adjustment of production levels to changes in demand
occurs quickly but the adjustment of output prices to changes in demand
occurs only gradually.
- g) A macro implication of this slow price adjustment is that shocks to 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 . | 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:*

- Exogenous Variables (determined outside the model):
- I, G, NE, a, b, t, with all terms strictly positive and b and t also strictly less than 1
- Endogenous Variables (determined within the model):
- C, Y

**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

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

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:*

- Exogenous Variables:
- I,G,a,g,m,b,t, with all terms strictly positive and with b and t also strictly less than 1.
- Endogenous Variables:
- Y^D, C^D, NE^D, Y, C, NE .

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

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 - mYupby 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.