The document summarizes the multiplier-accelerator model of economic fluctuations. The model combines the Keynesian multiplier, where investment is determined by changes in aggregate demand, and the accelerator theory, where investment is determined by changes in output. The model shows that if the accelerator coefficient is greater than 1, oscillations in output will be explosive, while if it is less than 1, oscillations will be damped. The model provides a simple explanation for cyclical fluctuations in economic activity over time.

1. 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).

2. 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

3. 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.

4. Example of the accelerator principle
a
Sherman & Kolk claim this is a reasonable figure since estimates show that GDP
is typically equal to 1/3 the value of the capital stock.
• We assume that ν = 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.

5. Time period Demand
for Shoes
Change in Demand
for Shoes
Shoe
Machinery
Change in Shoe
Machinery
1 $100 $300
1 to 2 $10 $30
2 $110 $330
2 to 3 $20 $60
3 $130 $390
3 to 4 $5 $15
4 $135 $405
4 to 5 $0 $0
5 $135 $405
5 to 6 -5 -$15
6 $130 $390

6. 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

7. Formalizing the model
•We assume that investment in the current period
(It) is equal to some fraction (ν) of change in
output in the previous period (or lagged output):
)( 21 −− −= ttt YYI ν (3)
•The consumption function is given by1
:
1−+= tt cYCC (2)
1
We assume that C depends on lagged, rather than current,
income. Also note that for our simplified economy, Y = YD.

8. Insert (2) and (3) into (1) to obtain:
21)( −− −++= ttt YYvcCY ν (4)
To get a homogenous equation, we ignore the
constant C
To get a standardized form, let A = c + ν.
Also, Let B = ν. Thus we can write:
021 =+− −− ttt BAY (5)
Note for the mathematically inclined: equation (5) is
a 2nd
order (homogenous) difference equation.

9. 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.

10. Period C Y Net I
1 $996
2 $1,000
3 $996 1000 $4
4 996 996 0
5 992 988 -4
6 985 977 -8
7 975 965 -11
8 964 952 -12
9 953 940 -13
10 942 930 -12
11 933 923 -10
12 927 920 -7
13 928 925 -3
14 928 933 5
15 936 944 8
16 945 956 11
17 957 969 12
18 969 982 13
19 978 991 13
20 987 996 9
21 992 1000 8
Example of the Multiplier-Accelerator
Assumptions: (1) Y is
$996 in period 1 and
$1000 in period 2;
(2) C = 96 + .9Yt - 1; and
(3) ν = 1

11. Multiplier-Accelerator Model
Data in Billions
Time Period
21191715131197531
NationalIncome(Y)
1020
1000
980
960
940
920
900
Assumptions: (1) Y is $996 in period 1 and $1000 in period 2;
(2) C = 996 + .9Yt -- 1; and (3) ν = B = 1

12. Time period
NationalIncome
Damped oscillations
B < 1 and A2
> 4B

13. Time period
NationalIncome
Explosive oscillations
B > 1 and A2
> 4B

14. 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.