A type of multiplier-accelerator model was employed by Lloyd Metzler (1941) in his famous "inventory cycle". Metzler's essential idea was that producers desire to keep inventory stock as some proportion of expected sales but, relying on lags between production and sales similar to that of Erik Lundberg (1937), Metzler contended that the precise inventory policy chosen by producers might have profound effects on the economy - particularly, in generating various different dynamics.
Let St be inventory stock. Sales are merely consumption, Ct. For simplicity, we can assume static adaptive expectations so that expected sales at time t are merely last period's sales, Cte = Ct-1. Thus, letting the desired amount of inventory at time t is St = kCt-1 where the parameter k reflects the proportion of expected sales that producers wish to held as inventory (we can presume that 0 < k < 1). Let us have a simple non-lagged consumption function where Ct = cYt. Thus:
St = kCt-1 = kcYt-1
is desired inventory at time t. However, recall that there might be inventory held over from last period as well. Namely, lagging everything once, St-1 = kCt-2 = kcYt-2 was the desired inventory level last period. There is no reason to suppose that this inventory was correct, thus expected sales last period Ct-1e = Ct-2 may have been different from actual sales Ct-1. Thus, the difference Ct-1 - Ct-2 = c(Yt-1 - Yt-2) reflects the unexpected discrepancy in sales from expectations and everything unsold (negative discrepancy) gets added to inventory while every item that was demanded in excess of expectations (positive discrepancy) gets depletes inventory. Thus, the inventories that result at the end of time t-1 are St-1 - c(Yt-1 - Yt-2), or intended inventories at t-1 plus/minus unexpected inventory accumulation/depletion. Thus:
St-1 - c(Yt-1 - Yt-2) = kcYt-2 - c(Yt-1 - Yt-2)
represents the inventory stock at the end of time t-1.
Now, total investment at any time t (call it It) is divided into increasing capital stock (ItK) and increasing inventory (ItS), thus It = ItK + ItS. For the sake of simplicity, let ItK be autonomous and constant, i.e. ItK = I0. Thus, total investment is merely investment into inventories. Producers investment in inventories at time t is the difference between desired inventory stocks at time t (St) and inventory stocks left over from time t-1 (as we saw, St-1 - c(Yt-1 - Yt-2)), or:
ItS = St - [St-1 - c(Yt-1 - Yt-2)]
Substituting in for St and St-1, then:
ItS = kcYt-1 - kcYt-2 + c(Yt-1 - Yt-2)
ItS = (1+ k)c(Yt-1 -Yt-2)
is thus our desired inventory investment at time t.
Now, output is produced to meet expected sales (Cte) and accumulate desired inventories (Its) and capital (ItK) thus:
Yt = Cte + ItS + Itk
(which is reminiscent but not the same as our old goods market equilibrium requirement - the difference being that we have firms's expectations of consumption rather than actual household consumption entering this equation). Thus, substituting our terms in:
Yt = cYt-1 + (1+ k)c(Yt-1 -Yt-2) + I0
Yt - c(2 + k)Yt-1 + c(1+k)Yt-2 = I0
which is a second-order linear difference equation very similar to the standard multiplier-accelerator model. The results are thus analogous. The particular solution (equilibrium) is merely:
Yp = I0/(1-c)
while the characteristic equation is:
r2 - (c(2+k))r + c(1+k) = 0
with the roots determined via:
r1, r2 = [c(2+k) ｱ ﾖ [(c(2+k))2 - 4c(1+k)]]/2
where we can derive different parameter regimes in (c, k) space in Figure 1. The disciminant is D = (c(2+k))2 - 4c(1+k) and so, for real roots (c(2+k))2 ｳ 4c(1+k) or, rearranging:
c(1 + k) + c ｳ 2ﾖ [c(1+k)]
1 + c(1 + k) - 2ﾖ [c(1+k)] ｳ 1 - c
thus we have either:
[1-ﾖ (c(1+k))]2 ｳ (1-c)
[1+ﾖ (c(1+k))]2 ｳ (1-c)
So, rewriting the first, 1 - ﾖ (c(1+k)) > ﾖ (1-c) thus, subtracting 1 from both sides, multiplying through by -1 and taking the square:
c(1+k) < (1 - ﾖ (1-c))2
k < [(1 - ﾖ (1-c))2 - c]/c
k < 2[(1-c) - ﾖ (1-c)]/c
which is the condition for damped monotonicity - normally, our area I dynamics in (k, c) space in Figure 1. However, as 0 ｣ c ｣ 1, then (1-c) < ﾖ (1-c), thus k must be negative. However, recall that k must be a positive value to make economic sense, thus damped monotonicity is not possible. However, doing the same exercise with the second condition, we obtain:
k > 2[(1-c) + ﾖ (1-c)]/c
for explosive monotonicty - which is entirely possible and corresponds to area II in Figure 1. Obviously, equality k = 2[(1-c) + ﾖ (1-c)]/c will obtain for constancy in area III. Thus, in Figure 1, we have drawn only the curve, k = 2[(1-c) + ﾖ (1-c)]/c. When k = 0, c = 1, thus there lies the maximum c. Explosive monotonic dynamics will be above this curve (area II), whereas constancy obtains everywhere along it (area III).
Figure 1 - Parameter regions for Metzler's Inventory Cycles
What about the complex roots case when (c(2 + k))2 < 4c(1+k)? Following the old modulus argument where now a = c(2 + k)/2 and b = (-D)/2, then:
a2 + b2 = c(2+k)/4 + [4c(1+k) - (c(2 + k))2]/4
a2 + b2 = c(1+k)
whereas ﾖ (a2 + b2) = ﾖ 1 = 1 = a2 + b2, then the critical function is c(1+k) = 1 or, as we draw it in Figure 1, k = (1-c)/c. Thus, we get unstable oscillations if k > (1-c)/c (area IV) and stable oscillations if k < (1-c)/c whereas we get constant oscillations when k = (1-c)/c (area V along curve).
Thus, Metzler's (1941) parameter regimes, as illustrated in Figure 1, are:
I - Damped monotonic (inoperable)
II - Constant monotonic
III - Explosive monotonic
IV - Explosive oscillations
V - Constant oscillations
VI - Damped oscillations
which shows the dynamic impact different inventory policies by firms (i.e. different values of k) can have on the qualitative dynamics of the economy.