Advanced Lecture Notes for Hall and Taylor, Chapter 6

Course Instructor: Professor Leigh Tesfatsion
Last Updated: 22 February 1996

HT6: Short-Run Fluctuations and Spending Balance

Basic References:  Hall and Taylor, Chapter 6;
                   Study Guide, Chapter 6.



#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

     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
              |                    .    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.


     Recall the form of the National Income Accounting Identity
for any specified time period T:

       Y  =  C   +   I   +    G   +   NE  ,


   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

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

       C^D   =   a  +  bY`d^e

            =   a  +  b[1-t]Y^e  ,

       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 = 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

        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 ,


 (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.


     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

     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

(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

   -$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

         q^0  +  q^1  +  q^2  +  ...    =   -----  ,

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

           dY^o             - bY^o
          ----   =      --------------     <     0  .
           dt            (1 - b[1-t])


     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.


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