Technical Appendix
For those who are comfortable with simple algebra, the
easiest way to develop the incomeexpenditure model is in
terms of equations. The consumption function we have been
using was:
(0) Consumption = $5000 + (3/4)(expected income)
The equilibrium condition we began with was:
(8) expected income = actual income
Since we began by assuming that the only form of spending
was consumption, and because spending is equal to actual
income because one person's spending is another's income, we
have this definition:
(9) actual income = consumption.
Using these three equations we obtain:
(10) expected income = $5000 + (3/4)(expected
income),
which yields, with a bit of algebra:
(11) (13/4)(expected income) = $5000
or:
(12) expected income = 4x$5000 = $20000.
Adding investment only slightly alters the equations that
represent the model. The equilibrium condition now includes
investment as part of actual income:
(13) Expected Income=Actual Income = Consumption +
Investment
The consumption function remains the same as above, but
the model now needs an equation to describe investment
behavior. The assumption used in the
example was:
(14) Investment = $2500
Substituting equations 0 and 14 into equation 13
gives:
(15) Expected Income = $5000 + (3/4)(Expected Income) +
$2500
or:
(16) (13/4)(expected income) = (1/4)(expected income) =
$5000 + $2500.
or:
(17) expected income = 4($7500) = $30000.
Notice that the multiplier of four shows up in equation
11 and that equilibrium income equals the multiplier times
the exogenous spending in the system (the $5000 plus the
$2500). You can see in equation 10 that the mpc determines
the size of the multiplier in this model. If, for example,
the mpc in equation 9 was 2/3 rather than 3/4, the
multiplier in equation 11 would be three rather than four.
In very simple models (but not in more complex models) the
multiplier will be 1/(1mpc).
The paradox of thrift can be demonstrated in this simple
model that excludes government using the
equation:
(7) S_{exp} + T = I + G
as the equilibrium condition. To solve for equilibrium,
we must find the savings function. Using the fact that
expected savings is the difference between expected
disposable income and consumption, and the consumption
function in equation 0, we can find the savings function for
our example as:
(18) S = Y  ($5000 + (3/4)(expected income))
or:
(19) S = 1/4(expected income)  $5000.
The onefourth in this equation is the marginal
propensity to save, the fraction of additional disposable
income that people will save.
If people become more thrifty, they save more and consume
less at each level of income. The 5000 in equations 0 and 18
becomes a smaller number. Suppose that it becomes 4000. Then
substituting our revised savings function and the investment
equation 14 into equation 7, we get:
(20) I+G = $2500 = (1/4)(expected income)  $4000 =
S+T
or:
(20) $6500 = 1/4(expected income)
or:
(21) $26000 = expected income.
Comparing the new equilibrium income to the old one, we
find that an extra $1000 of thriftiness results in a $4000
drop in incomethe multiplier was at work again. Savings
remain at $2500.
Adding taxes complicates the consumption function. Taxes
(abbreviated T) will create a gap between the total income
of the nation and the disposable income that people can
spend or save. Thus the consumption function is changed,
with the revised consumption function assuming that
consumption depends on expected aftertax income, or:
(22) C = a + (mpc)(expected income  taxes)
where the "a" term is the intercept. (It took the value
of $5000 in equation 0.) Two other equations must be added
to the system to explain the behavior of government. The
simplest assumption for both government spending and taxes
is that they are determined outside the system as policy
variables, or:
(23) G = g
(24) T = t
The solution to this revised set of equations is
straightforward. First, recall that the equilibrium
condition is that expected income equal the sum of
consumption, investment, and government spending. Then
suppose that government spending and taxes are both $1000,
that investment is fixed at $2500 as before, and that the
intercept and mpc in the consumption function remain $5000
and 3/4, respectively. The substituting into the equilibrium
condition, we get:
(25) expected income = $5000 + (3/4)(expected income
1000)+2500+1000
or:
(26) expected income = 4($5000 + $2500 +$1000)  3($1000)
= $31000.
You might have noted that the way we divide spending into
parts is based on the way GDP can be split into its uses.
The incomeexpenditure model tries to determine the size of
each slice of GNP and then adds them together to discover
the whole. We have not up to now included net exports, and
for completeness they must be included. Exports act as an
injection of spending into the system, while imports act as
a leakage of spending from the system. Many textbook models
eliminated net exports by assuming that the economy is
"closed," which means that it has no imports or exports.
Finally, you might also notice the similarity of equation
7 to the budget constraint for the economy, which said
that
(27) (IS) + (G + Transfers  T) + Net Exports = 0
In the simple incomeexpenditure model, the only one of
these variables that can adjust is savings, and the only way
it can adjust is through a change in income. Thus if
investment or government spending or net exports fall, the
incomeexpenditure model asserts that income must also fall,
because this is the only way it allows for savings to
change. No modern economists believes a model this
simplistic; they give some roles to changes in interest
rates and prices. But those economists who believe that the
incomeexpenditure model captures an important part of the
story stress the adjustments in savings caused by changes in
income.
Copyright
Robert Schenk
