The multiplier-accelerator model
Initial points
1. The model is a synthesis of the Kahn-Keynes
multiplier and the “accelerator” theory of
investment1.
2. The accelerator model is based on the truism that, if
technology (and thus the capital/output ratio) is
held constant, an increase in output can only be
achieved though an increase in the capital stock.
P. Samuelson. “Interaction Between the Multiplier Analysis and the Principle of
Acceleration,” Review of Economic Statistics (May 1939).
The accelerator
•Firms need a given quantity of capital to produce the
current level of output. If the level of output changes, they
will need more capital. How much more?
Change in capital = accelerator  change in output (10.1)
•But firms can only increase their capital stock by
(positive) net investment. How much?
Net investment = accelerator  change in output (10.2)
•It is also true that:
Accelerator = Change in Capital/Change in Output
Capital/Output ratio
•If we do not allow for productivity
boosting technical change, then the
capital output ratio is held constant.
•If fact, this is what we are
assuming—no technical change.
Example of the accelerator principle
• We assume that  = 3a . That is, it takes 3 dollars
worth of capital to manufacture $1 worth of shoes.
•Hence if the demand for shoes increased by say, $10,
there would be a need for $30 in additional capital—or
equivalently, $30 in net investment.
aSherman & Kolk claim this is a reasonable figure since estimates show that GDP
is typically equal to 1/3 the value of the capital stock.
Time period
1
Demand Change in Demand
for Shoes
for Shoes
$100
1 to 2
2
$110
4
$60
$390
$5
$135
$15
$405
$0
$135
5 to 6
6
$330
$130
4 to 5
5
$30
$20
3 to 4
$0
$405
-5
$130
Change in Shoe
Machinery
$300
$10
2 to 3
3
Shoe
Machinery
-$15
$390
Formalizing the model
If the economy is in equilibrium,
Then output supplied (Y) is equal to
aggregate demand (AD). Assuming a
closed economy without
government, we have:
Yt = Ct + It
(1)
Formalizing the model
•The consumption function is given by1:
Ct  C  cYt  1
(2)
•We assume that investment in the current period
(It) is equal to some fraction () of change in
output in the previous period (or lagged output):
It   (Yt  1  Yt  2)
(3)
We assume that C depends on lagged, rather than current,
income. Also note that for our simplified economy, Y = YD.
1
Insert (2) and (3) into (1) to obtain:
Yt  C  (c  v)Yt  1 Yt  2
(4)
To get a homogenous equation, we ignore the
constant C
To get a standardized form, let A = c + .
Also, Let B = . Thus we can write:
Yt  At  1  Bt  2  0
(5)
Note for the mathematically inclined: equation (5) is
a 2nd order (homogenous) difference equation.
It can be shown that:
1. There will be cyclical fluctuations in the time
path of national income (Yt) if A2 < 4B.
2. If B = 1 (and presuming that A2 < 4B), then
cycles are constant in amplitude.
3. If B < 1 (and presuming that A2 < 4B), then
cycles are damped—that is, amplitude is a
decreasing function of time.
4. If B > 1 (and presuming that A2 < 4B), then
cycles are explosive—that is, amplitude is a
increasing function of time.
5. There will be no cyclical fluctuations if A2 >
4B.
Example of the Multiplier-Accelerator
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
C
$996
996
992
985
975
964
953
942
933
927
928
928
936
945
957
969
978
987
992
Y
Net I
$996
$1,000
1000
$4
996
0
988
-4
977
-8
965
-11
952
-12
940
-13
930
-12
923
-10
920
-7
925
-3
933
5
944
8
956
11
969
12
982
13
991
13
996
9
1000
8
Assumptions: (1) Y is
$996 in period 1 and
$1000 in period 2;
(2) C = 96 + .9Yt - 1; and
(3)  = 1
Multiplier-Accel erator Model
Assumptions: (1) Y is $996 in period 1 and $1000 in period 2;
(2) C = 996 + .9Yt -- 1; and (3)  = B = 1
Data in Billions
1020
1000
980
960
940
920
900
1
3
5
7
9
11
13
Time Period
15
17
19
21
Damped oscillations
B < 1 and A2 > 4B
Time period
Explosive oscillations
B > 1 and A2 > 4B
Time period
Qualifications/limitations
•This model is based on a crude theory of investment.
There is no role for “expected profits” or “animal
spirits.”
•The time lag between a change in output and a
change in (net) investment can be significant—the
investment process (planning, finance, procurement,
manufacturing, installation, training) is often lengthy.
•J. Hicks pointed out that, for the economy as a
whole, there is a limit to disinvestment (negative net
investment). At the aggregate level, the limit to
capital reduction in a given period is the wear and
tear due to depreciation.