One may argue that the components of Hicks's (1950)
growth cycle model, ceilings, floors and an exogenous rate of growth of autonomous
investment (equal to the rate of growth of factor supplies) is unsatisfactory. These
exogenous factors are, indeed, quite arbitrary and, as their values can be anywhere, there
does not seem to *essential* cycles. A way to circumvent this is to appeal to
theories which obtain growth and cycles endogenously. Non-linear cycle models were
developed precisely for this purpose. However, the simplest alternative procedure to
obtain growth and cycles endogenously would be to modify the Hicks model with Duesenberry
"ratchet" effects on consumption and/or Smithies "ratchet" effects on
investment - more complicated non-linear models being left for later.

James Duesenberry (1949) proposes a consumption function which accounts for some adjustments for "habits" or "standards of living". Conventionally, if income falls, then consumption should fall proportionally with the marginal propensity to consume. Duesenberry rejected this: once consumption habits are acquired, it is hard to get rid of them. Thus, income shocks should have slightly different effects on consumption. Certain consumption habits are formed at high income levels which are not completely abandoned when income falls. This effect is captured in the following consumption function:

C

_{t}= c_{0}+ c_{1}Y_{t-1}+ c_{2}Y_{t-1}^{m}

where Y_{t-1}^{m} is the maximum level of income before
time period t. Thus, consumption habits acquired when income was at its highest influence
present consumption decisions.

Unlike Duesenberry, Arthur Smithies's (1957) "ratchet" effects are actually applied to the investment function rather than to the consumption function and they also end up yielding cycles around a growth trend. However, we shall stick mainly to the Duesenberry consumption function and omit the Smithies ratchet - thus we presume the investment function has the same presumed to have the same properties as before (now without depreciation) so:

I

_{t}= I_{0}e^{gt}+ b (Y_{t-1}- Y_{t-2})

thus, for goods market equilibrium:

Y

_{t}= c_{0}+ I_{0}e^{gt}+ c_{1}Y_{t-1}+ c_{2}Y_{t-1}^{m}+ b (Y_{t-1}- Y_{t-2})

or:

Y

_{t}- (c_{1}+ b )Y_{t-1}+ b Y_{t-2}= (c_{0}+ I_{0}e^{gt}+ c_{2}Y_{t-1}^{m})

which is our old Hicksian second order linear
difference equation - only now with the term c_{2}Y_{t-1}^{m}
involved among its autonomous components.

Without looking too carefully, one might suspect that the only effect
would be then that the particular numerical value of the equilibrium level of output would
be different. But in fact, the effect of adding a "ratchet" to consumption is
more substantial. Namely, we have changing parameters and changing steady-state
equilibria. This is easy to see. Suppose that Y_{t-1}^{m} is far in the
past - call this case 1. In this case, then, the particular solution (equilibrium) to the
difference equation is:

Y

^{1*}= (c_{0}+ I_{0}e^{gt}+ c_{2}I_{0}e^{gt})/(1-c_{1})

where, assuming that c_{1} = c (our old mpc), our characteristic
equation is now:

r

^{2}- (c_{1}+ b )r + b = 0

then we can just draw this as in the old Hicksian (b
, c_{1}) space as in Figure 1 below. The only difference is that our curves are
now b = (1 - ﾖ (1-c_{1}))^{2}
and b = (1 + ﾖ (1-c_{1}))^{2}
where again monotonicity is defined when b is above this and
oscillation when it is below and the border between stable and unstable dynamics remains b = 1. Thus, nothing substantial changes except the numerical value
of the equilibrium.

But now suppose that an economy is growing. Then, in this case, call it
case 2, Y_{t-1}^{m} = Y_{t-1}, i.e. last period's output __is__
the maximum output. Thus, for case 2, the difference equation becomes:

Y

_{t}- (c_{1}+ c_{2}+ b )Y_{t-1}+ b Y_{t-2}= (c_{0}+ I_{0}e^{gt})

so that the particular solution (equilibrium) is:

Y

^{*2}= (c_{0}+ I_{0}e^{gt})/(1-c_{1}-c_{2})

which is different from that of case 1 as now c_{2} enters in the
denominator as part of the multiplier. Similarly, under case 2, the characteristic
equation becomes:

r

^{2}- (c_{1}+ c_{2}+ b )r + b = 0

which is very different from case 1. It is easy to see that the curves for
the parameter constellations in (b , c_{1}) space are
now:

b = (1 + ﾖ (1-c

_{1}- c_{2}))^{2}

and:

b = (1 - ﾖ (1-c

_{1}- c_{2}))^{2}

which are different from case 1. Thus, the boundaries we had before now
shift "in" as shown in Figure 1. When b = 1,
now, c_{1} = 1 - c_{2} and not 1.

Figure 1- Parameter Regions with Duesenberry Ratchet Effects

What are the implications of this? Well, quite simply, the properties of
the parameters change during the cycle - thus one may consider it somewhat
"non-linear". Let us consider two cases: that of points P and Q in the figure
above. For the sake of simplicity, we shall assume henceforth that the exogenous rate of
growth for autonomous investment is zero, i.e. g = 0 so that I_{0}e^{gt} =
I_{0}. Thus, our equilibria Y^{*} are constant.

Suppose we have specific values of b and c_{1}
(assumed constant) at point P. Then we alternate between explosive oscillations to
explosive monotonic dynamics. This is not altogether very interesting. The situation is
illustrated in Figure 2. Suppose we begin at a time when Y_{t-1}^{m} is
max_{1}. Then, from t = 0 to t = T, the maximum is long in the past (max_{1})
and our equilibrium is Y_{1}^{*}. Under this, then, we have case 1 and
thus explosive oscillations. So, as we begin from time t = 0, Y has explosive
oscillations.

Figure 2-Explosive Oscillations to Monotonicity

However, eventually, as the swings of Y are growing, we shall get a value
of Y which is greater than max_{1}. In our case, Y achieves and surpasses max_{1}
at t = T (or rather, just before that) in Figure 2. As a result, case 2 comes into effect.
Now, we have two things: firstly, a new, higher equilibrium (Y_{2}^{*})
will result and a new higher maximum will be established corresponding to the Y achieved
at t = T. However, as we now have explosive monotonic dynamics, then Y will *not*
move down towards the new equilibrium Y_{2}^{*}. Instead, it will continue
to explode upwards. But as it explodes, it will create a new higher maximum Y and thus a
new higher equilibrium Y^{*} after T. But the explosive monotonicity maintains
itself so again, the maximum continues to grow and Y^{*} continues to grow. This
is shown by the two lines Y* and max Y after t = T in the figure above.

The case of P, illustrated in Figure 2, is not that interesting for
we move from oscillations into monotonic explosions which eliminates the cycle. However,
suppose we were with a parameter constellation such as Q in Figure 1. Then, *both*
before and after the parameter changes, we would *still* have explosive oscillations.
This case is illustrated in Figure 3 below. We begin, again, at t = 0 with a maximum Y (=
max_{1}) and equilibrium Y_{1}^{*}. We have explosive oscillations
and these continue until we hit t = T_{1}. Then, our upward swing of Y moves above
max_{1} all the way to max_{2}. This becomes the new maximum and, as a
result, the new equilibrium is Y_{2}^{*}. However, given that we still
have explosive oscillations, output falls after t = T_{1} and, indeed, falls below
the new equilibrium. But, as we have explosive oscillations, then sometime in the next
upswing, at t = T_{2}, Y rises above max_{2} all the way to max_{3}.
This creates a new maximum (max_{3}) and a new, higher equilibrium Y_{3}^{*}.
Again, because we have oscillations, then from t = T_{2} to t = T_{3}, the
rest of the cycle works itself through normally until, of course, during the upswing, at t
= T_{3}, again we surpass the old maximum to obtain a new maximum (max_{4})
and a new higher equilibrium, Y_{4}^{*}.

Figure 3- CyclesandGrowth

We can see now what is happening in Figure 3. The equilibrium Y^{*}
is rising stepwise and oscillations are happening around this equilibrium. Obviously,
then, we have Harrod's grail of growth *with *cycles:
Y^{*} increases while actual Y = Y_{t} is moving in cycles along this
increasing equilibrium. This is the result of parameter constellations such as Q when you
add Duesenberry (1949) "ratchet"
effects on consumption on the simple Hicksian
multiplier-accelerator model: cyclical growth without once the need for the Hicksian
paraphenelia of exogenous growth rates, ceilings or floors. If we had assumed a positive
exogenous growth rate (g > 0), then the equilibrium Y^{*} would be steeper
(i.e. the steps would have been greater) - perhaps to a point where the downswing would
result in only slightly or even no negative growth of output. Amplitudes of the cycles, of
course, should not be assumed constant but probably increasing over time.

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