David K.H. Begg, Gianluigi Vernasca-Economics-McGraw Hill Higher Education (2011)

CHAPTER 27 Business cycles
The multiplier-accelerator model explains business cycles by the dynamic interaction of consumption and
investment demand. The insight of the model is that it takes accelerating output growth to increase
investment. Once output growth stabilizes, so does investment. In the following period, investment must
fall, since output growth has been reduced. The economy moves into a period of recession, but once the
rate of output fall stops accelerating, investment starts to pick up again.
This simple model is not the definitive model of a business cycle. If output keeps cycling, surely firms stop
extrapolating past output growth to form assessments of future profits? Firms, like economists, recognize
that there is a business cycle. The less investment decisions respond to the most recent change in past
output, the less pronounced will be the cycle.
The multiplier-accelerator model of cycles
II
Suppose I denotes current investment, L1 denotes investment in the previous period, Y denotes
output and dY denotes (Y- Y_1 ), the increase in output between last period and the current
period. Output Y is related to current investment I by the multiplier Y = 1/(1 - c), where c is the marginal
propensity to consume. Investment depends on output growth, so I= a d Y. Hence,
Hence
I = a d Y = [al( 1 - c)] [I
- I_ 1]
I= -{ a/[1 - c - a]}l_1
This equation is of the general form I= bL1• If b is a positive fraction, I is always smaller than the period
before and gradually converges on zero. If b exceeds unity, I gets larger and larger for ever. Negative values of
b imply I becomes negative every second period, either converging to zero or becoming ever larger. None of
this generates things like business cycles.
Cycles emerge however with small changes to these formulae. Table 27.1 offers one example. Here is another.
Suppose the consumption function depends not on current income but on income the previous period
so that C = A + c Y_1 and current investment depends on output growth in the previous period, so that I =
a[Y_1 - Y_2]. Since Y = C +I in this simple economy,
If the economy is in long-run equilibrium, output is constant, the final term is zero, and equilibrium output
Y* is given by Y*= Al(I - c). Usingy to denote Y - Y*, the deviation of output from its long-run level, we can
subtract Y* from both sides of equation ( 1) to yield
y = cy_1 + a[y_1 - y_2] (2)
Depending on the values of c and a, equation (2) can yield constant cycles, damped cycles that gradually get
smaller and smaller, or explosive cycles that get larger and larger. When c = a, we simply get
(y - Y-1 ) = -(y_. - Y- 2 )
so that positive and negative growth of similar size alternate for ever.
(1)
Ceilings and floors
The multiplier-accelerator model can generate cycles even without any physical limits on the extent of
fluctuations. Cycles are even more likely when we recognize the limits imposed by supply and demand.
Aggregate supply provides a ceiling in practice. Although it is possible temporarily to meet high aggregate
demand by working overtime and running down stocks of finished goods, output cannot expand indefinitely.
620