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Inventories in a dynamic macro model with flexible prices
European Economic Review 27 (1985) 201-219. North-Holland
INVENTORIES
IN A DYNAMIC MACRO FLEXIBLE PRICES*
MODEL
WITH
John C. ECKALBAR California
State University,
Chico,
California
95926,
CA
Received June 1984, tinal version received November 1984 This paper presents a disequilibrium macro model with inventories, rational expectations, and flexible prices. It is found that less than full employment equilibria are possible, even highly likely, in the intermediate term and that wage reductions will cause employment to drop. Increasing aggregate demand will raise employment only if it is done gradually. On a technical level, we have multiple switching lines, disconnected regimes, and a connected equilibrium set which borders on a switching ljne and otherwise lies in a single region.
1. Introduction Just a few years ago, there were hardly any theoretical macro models with inventories.’ In retrospect this is hard to understand in view of the fact that economists, at least since Keynes, ’ have felt that inventories play a key role in macro dynamics, and statistical studies have consistently shown that inventory fluctuations account for a substantial fraction of cyclical output variations. Now, thanks largely to a series of papers by Alan Blinder, we are seeing a burst of interest in inventory theoretic macro modeling. Virtually every major segment of macro theory has been influenced - there are IS-LM based models [Blinder (1977,1980)], rational expectations models [Blinder and Fischer (1981), Green and Laffont (1981), Brunner, Cukierman and Meltzer (1980)], and disequilibrium models [Honkapohja and Ito (1980), Eckalbar (1985), Simonovits (1982)].3 If there is a common weakness to these models as a group, it is their treatment of prices. Existing models seem to either treat prices as fixed, sometimes ignoring them altogether [Honkapohja and Ito (1980), Eckalbar 71985), Simonovits (1982), Chaudbury (1979), Blinder (1977)], or they assume that prices adjust instantly to equate demand and *Research reported here was partly funded by a grant from the National Science Foundation. ‘Exceptions: Metzler (1941), Love1 (1961). ‘The General Theory of Employment, Interest and Money (1936, p. 318). % addition, there is considerable work under way on the microeconomics of inventories: Reagan (1982), Amihud and Mendelson (1982, 1983). 00142921/85/%3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)
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supply, or the expected values of demand and supply [Blinder and Fischer (1981), Brunner, Cukierman and Meltzer (1980), Green and Laffont (1981)]. The present paper incorporates flexible prices, without resorting to the unrealistic assumption of continuous market clearing. The firm is assumed to be a price setter. It has a fixed target inventory-expected sales ratio; and it raises (lowers) price whenever actual inventories are below (above) their target level. It is found that in the ‘intermediate term’, i.e., while nominal wages are fixed, less than full employment equilibria are possible, even though real wages are flexible. In the long run, (i) wage reductions are likely to lower the employment level; and (ii) a ‘moderate’ program of demand expansion can lead to full employment, while an ‘excessive’ boost in demand can be counterproductive.4 On the technical side, the model presented here has several novel features. (i) There are multiple switching lines, which do not intersect at an equilibrium. (ii) The equilibrium is a connected set. (iii) There are branching points, or points of indecision in the phase space. (iv) Individual regime domains are disconnected. Section 2 gives the model. Section 3 outlines the comparative statics, and section 4 is the conclusion. 2. The model
The model contains two goods, labor and output, and two sectors, households and firms. The household offers a fixed quantity of labor, L, each period and buys output according to the sales equation, a+cwL
s=p,
(1)
where a and c are constants (a>O; 0
CM, + CWL P
’
and ‘excessive’ are given precise definitions in the text,
(2)
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where M, is beginning period money holdings. If the household always receives dividends equal to profits, M, is constant.’ Then, since we don’t have any pretense to a monetary theory here, we will set CM, = a to yield (1). The firm is responsible for hiring labor, producing output, making sales, and adjusting prices. We assume that the firm forms expectations of upcoming sales and, if it is profitable to do so, tries to hire enough labor to provide for these sales and leave itself with inventory, K equal to kS’, where k is a positive constant and Se is the expected level of sales. Of all the simple assumptions one could make about target inventory levels, the fixed target ratio to sales seems most defensible. The actual ratio of inventories to sales is trendless over the past forty years, and cyclical turns in the ratio can be accounted for by slowly adjusting expectations.6 For simplicity, we assume that output, Q, is given by Q=dL,
d>O.
(3)
If w/p>d, production is unprofitable, and labor demand will be zero. But if w/p 5 d, the firm will wayt to produce Q=max[(l+k)S’-KO].
(4)
Two remarks: (i) (1 + k)Se- V is the output level which would provide for all anticipated sales and bring inventory to its target level, kS’. We are implicitly assuming that the firm makes an effort to adjust inventory completely within the period. Feldstein and Auerbach offer empirical support for this assumption. (ii) If V is high relative to S”, negative output would be called for were it not for the max condition in (4). The above considerations give rise to a labor demand equation of the following form: Ld=min($-d,O)*[l/f-d]max[(l+k)S’-KO]*-$ Note that high inventory levels tend to reduce desired output and derived labor demand, as most micro theoretic discussions have shown, and note that d Ld/d (w/p) s 0, with other variables constant. As is usual in this class of models, we assume that the quantity of labor actually hired is the lesser of the quantities supplied and demanded, Thus: L = min (L, Ld),
and the firm may be unable to produce everything it wishes. ‘See my ‘Stable Quantities . . .’ for an alternative formulation. %ee Eckalbar (1985).
(6)
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The dynamics of the model are as follows: Wages, prices, inventories, and sales expectations give rise to Ld, then min(Ld, L) gives L, which determines S and Q. The values for S and Q imply the change in inventories, Q-S. The firm is assumed to be a price setter. We will suppose that the firm compares desired inventories, kS”, to actual inventories, r! and adjusts prices upward (downward) if kS’ is greater (less) than K Recent papers by Reagan (1982), Blinder (1982), and Amihud and Mendelson (1982,1983) offer a solid dynamic programming microfoundation for our price adjustment assumption. If we assume that wages adjust according to the excess demand in the labor market and that sales expectations change according to an as yet unspecified function f( ), we have the following dynamical system: ti=h(Ld-L), fi = g( kS” - I’), (1)
P = d. min (Ld, L) - [a + cw min (Ld, L)]/p,
where Ld is given by (5) and h() and g() are sign preserving. System (I) has three switching surfaces [Ld =L, w/p=d, and (1 + k)Se= P’l in four dimensions. A model of this complexity is, at least at the present, beyond the range of existing disequilibrium techniques.’ However, two commonly used assumptions are sufficient to render the system tractable while still preserving enough of the ‘detail to keep the problem interesting. First, we assume that w is fixed in the short run. Later, if we find that the system settles at a less than full employment equilibrium, we will consider parametric changes in w - but for the moment, w will be thought of as infinitely slow relative to the remaining variables. Second, we assume that sales expectations are formed ‘rationally’. In the present context this means that if the firm hires labor on the assumption that sales will equal Se, sales will in fact equal S,, when that quantity of labor is hired. In most works of this sort, ‘rationality’ is a matter of faith. But if a theorist cannot derive rationality from some defensible learning process, then the assumption seems unwarranted. Accordingly, we digress for a moment to consider learning. Following Benjamin Friedman, we take adaptive expectations to be an approximation of optimal least-squares learning. Imagine that with p and V fixed, households and firms go through tatonnement with respect to S and Ld. With S’ adjusting adaptively, we have the following system for the region ‘See Honkapohja and Ito (1983), Eckalbar (1980).
J.C.
where O
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CL.
S,e=SC-l+a(S,_,-S:-,),
W
in dynamic
L:+l+k)s:-V-J,
O
a+CW,L; P .
This yields the first order difference equation (7) which is convergent if p>cw,(l +k)/d. This restriction will be examined shortly. In the regions where Ld = 0 and L=L, the difference equation becomes SF= (1 - a)S:- I + constant, so no additional restriction is necessary here to guarantee convergence to ‘rationality’. This sort of no-trade tatonnement in S and Ld (‘tell me what you’re going to do and I’ll tell you what I’m going to do’) is not a completely satisfactory characterization of the learning process. As an alternative, we might suppose that prices and wages are fixed, but that trade actually takes place so that V and S’ both change during learning. Assuming continuous adjustment, we have the following expectation adjustment system:
(EJ
Se=a(S-Se),
v=Q-S.
In the region where 0 < Ld -CL, we have:
where p and P are deviations from equilibria, and H is a column vector of constants. The Routh-Hurwitz conditions show that the learning process in (E;) is convergent if p> [cw,( 1 + k) -cwe]/( 1 +a)d. As a varies from 0 to 1, the lower limit for p varies from cw,/d to cw,(l + Q/d. A comment on this shortly. For the regions L=O and L= E, no additional restrictions are required to yield a convergent learning process.
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Notice that convergence to ‘rational’ expectations is problematic only in the region where 0 < Ldd/c(l -t k). But when the real wage is high enough, Ld=O, and it may be that if 0~ Ld
ka + kb min (Ld, L) _ I/ P
,
where for simplicity we have defined b =cwo, and we will now take g to be a positive constant. Our method will be to investigate this system one regime at a time and then patch the various regimes together for a look at the whole system. First, consider R,={(p, v)) w,/d>p}. In RI the real wage exceeds the marginal product of labor, and profitable production is impossible. It follows from (5) that Ld=O in RI; thus in this subspace (II) becomes (III -3) V= -$
;-Jf.
Ij=g
,
(
>
This gives rise to the flow depicted’in fig. 1. Note that V is falling everywhere in R,, and that p is falling (rising) everywhere above (below) the locus A, = {(p, v) 1V= ka/p}. The latter fact rules out negative prices. If p> we/d, then by (5) we have Ld=fmax[(l If O
+k)S-
KO].
(8)
we have
1 .
Solving for Ld using rational expectations, we have L,js(l+k)a-PV dp-(1 +k)b’
Note that (10) is discontinuous
O
at p = (1 + k)b/d.
(9)
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d
Fig. 1
Defining T = {(p, the equation
V) 1 Ld = E},
we find that the switching line T is given by
The locus T is asymptotic to the vertical axis and to the line V= -dL. Checking (10) we see that (where it exists)
aP
-P
z=
dp-(1 +k)b’
(12)
hence a small increase in V from a point on T will yield Ld (1 + k) b/d, and vice versa. Thus, the lines p=(l + k)b/d and T further divide the phase space into four general regions. See fig. 2. In R, and R, we are either above T and right of
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r
K
d \
T
Fig. 2
p=( 1 + k)b/d or below T and left of p=( 1 +k)b/d. Given (12), we know that Ld w,/d, Ld =O}. Looking back to (12) we see that the full employment regime, R4 = {(p, V) ) Ld > L = L}. Each of these three new regimes has its own dynamical system. Starting with the simplest case, the flow in R, is given by (II, -J above. Inventories are falling everywhere in R,, and prices are falling (rising) everywhere above (below) the locus A,. Note that no rest point is possible in either region R, or R3. The next simplest case is that of full employment in R,. In this case (II) becomes
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v=
(j-b (
jQ! P>
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model
+g[~(a+bwj.
System(I1,) defines two loci, pi= {(p, v) ERq/ i’=O} Points in $‘i are on the vertical line given by
and @z= {(p, v) ER,l/j=O}.
bZ+a P=dL.
Everywhere in R, to the left (right) of this line inventories are falling (rising). The locus i/z may fall on either side of the line p =(l + k)b/d. In order to keep our figure drawing to a minimum, we will assume that (bL+ a)/dL> (1 + k)b/d and leave it as an exercise for the reader to redraw the figures for the other case. The locus @i is a rectangular hyperbola with both axes as asymptotes, it is given by the line
V=k(a+bf) P
(14)
’
The loci J$ and vz meet on T (A.l)* at the full employment
equilibrium
(15)
.
The locus iz cuts T from below, passing through R, into R, as shown in fig. 3 (A.2). (Here and elsewhere, a locus will be shown as a dashed line over any portion of itself outside its domain.) It is clear from inspection of (II,) that p is falling everywhere in R, above pz, and vice versa. This produces the vector field shown in fig. 3. If the full employment equilibrium were in the interior of Rq, it would be locally asymptotically stable (A.3). But we know from (A.l) that (p*, I’*) falls on the border between R, and R,, so any stability check is premature at this point. The only region remaining is R2, within which 0 -CL* = L < L. Substituting (10) for L* in (II) we have
1-a
p’
‘To preserve the flow of the text, numerous minor assertions will be proven in the appendix and marked (A.l), (A.2), etc.
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-! P: V’) ---_ R2
A+
\
Fig.
Defining ti given by v=
and fig analogous with vi and rji, we find that both loci are kad dp-b’
(16)
The implication is that any point in R, satisfying (16) is a less than full employment equilibrium. But does I’!#) run through R,? Yes. k’r&$) is a rectangular hyperbole with asymptotes at p= b/d and V=O. It intersects T at exactly two points - (p*, V*) and (b(1 + k)/d, ad/b) (A.4). At (p*, V*) the locus i/8(@8) is flatter than T; while at (b(1 + k)/d, ad/b), it is steeper than T (A.5). This implies that @(pi) passes into one section of R, as shown in fig. 4. The significance of this is that the complete system, (II), does not have a unique equilibrium - it has an equilibrium locus, with a full employment equilibrium as an endpoint and an infinity of less than full employment equilibria trailing downward to the right. For convenience, we define the equilibrium set E = {(p, V)l(p, V) ER,, V = kad/(dp - b)}.
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a+bi dt Fig. 4
We are now in a position to complete the dynamic picture of (II) by in the flow in R,. From (II,), we see that
aif
-(dp-b)
%=dp-(1
+k)b’
(17)
Since p > b/d in R,, (17) is positive if p < (1 + k)b/d and negative otherwise.g ‘Note the frequency that the term dp-( 1+ k)b has come up in the text. If p>( I+ k)b/d, optimal OLS learning will always converge, p=( 1+ k)b/d constitutes a regime border, etc. In Honkapohja and Ito (1980) and Simonovitz (1982), where prices are suppressed, an equivalent term (a-b(1 +p)) proves to be critical, but they are forced to simply assume that it takes one sign or another. [See Honkapohja-Ito (1980, p. 189).] Having prices variable is clearly superior in that: (i) We don’t have to restrict ourselves to one sign for dp-( 1 +k)b; as it changes sign, we simply switch regions. (ii) We are able to shed some light on why dp-( 1 + k)b is critical, i.e., in terms of learning and regime switching.
E.E.R.-
D
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This means that in R, near E, V is decreasing increasing if it is below. See fig. 4. A similar argument would show that
model
if it is above @ and
ati -d&-b)
W
a=+(l+k)b
which gives the flow direction for p in fig. 4. Figs. 14 can be combined to give In the interest of holding down the dispense with the variety of switching of the flow. This is done in fig. 5.
R, on either side of t!&ji).
Again, see
a complete picture of the flow of (II). complexity of the resulting figure, we loci and show only the general nature
Three technical notes: First, on the line p=( 1+ k)b/d, between A, and fii, the system has ‘branching points’ or ‘points of indecision’.” That is, points slightly right of VI
I
P
0
Fig. 5 ‘OSee Flugge-Lotz.
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this segment move down and right (into R4), ‘while points slightly left move down and left (into R3). The problem is trivial, since no path moves onto this segment, and any displacement from the segment leads to a fully determinant path. Second, nothing in the analysis precludes inventories from becoming negative. This might be dealt with in two ways: (i) we might impose a border restriction on t such as i/z0 for T/=0; or (ii) we might let V become negative and think of V as simply a backlog and S as an order rather than a sale. The latter is clearly a better solution, but can that upset the stability of E? No. if we cross below V=O, it will have to be at some p <(a +bE)/dL, but then p will always be increasing along such a path, so we must sooner or later find p > (a+ bE)/dL. If so, V will begin increasing and will sooner or later cross V=O again. Hence, V is bounded below along any path. Prices could never become negative, since 1ir1+~ S= 03, and price change is positively linked to S. See (II, -3). Third, the regions where labor demand and output are zero (R, and R3) are included for analytical,completeness. They are largely a byproduct of the assumption of linear technology, and they are of little economic relevance. Note that these regions play no role in the local dynamics around E, which is contained in R, and R,. Finally, the stability of E can be rigorously established with a proof which follows the method of theorem 5 of my Econometrica paper, but in this case the proof will be dispensed with, since stability of E is obvious from visual inspection. 3. The comparative
statics
(i) We have been supposing that w is fixed as the system makes its way to E, i.e., that wages are infinitely slow. But what if we settle at some unemployment equilibrium, and then the wage falls? Does the wage adjustment move the system closer to full employment? Until now the wage has been subsumed into b via b=cw,. So a reduction in the wage translates into a drop in b. Looking back at the figures, we see that a reduction in b slides the equilibrium locus, along with every other reference line, to the left. So if we began at an equilibrium like point x in fig. 6, a one time drop in w would lead us to a point on E’ somewhere between y and z. Notice that V will certainly be lower at the new equilibrium. This is significant. It can be shown (A.6) that on E, employment and labor demand are given by L = Ld = V/kd.
Thus, equilibrium
employment
(1%
is, a fixed multiple
of the equilibrium
inven-
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V
V’kdi
c1
P
Fig. 6
tory level. It follows from this that employment is less along the segment yz than at x, or that the wage reduction makes matters worse in the labor market. The reason for this is that when wages drop, the household’s purchasing power also drops; and the firm’s labor demand is predicated upon sales to households. So as households buy less, firms hire fewer workers. (ii) Thus far we have not built in a government sector, but it is easy to see how that might be done. We have already shown that autonomous sales, a, might be taken to be cMo, thus an increase in the supply of money would transalte into an increase .in a. Similarly, since a is simply nominal autonomous sales demand, an increase in government demand would be equivalent to an increase in a. When a increases, most of the loci, including E, shift to the right. If such a policy is undertaken when the system is initially at a less than full employment equilibrium, then after the shift in E, we would be at an ‘initial’ point somewhere in R, or R4 below V= kdz. We can be sure that an increase in u will cause prices to rise, but the employment outcome is less certain. If the shift in E leaves our initial point
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somewhere in R, or in R, above P=(a’+bt)/dL (where a’ is the new higher level of a), then I/ will be everywhere increasing along the path to E’, and we must end up with V and, therefore, L above their original values. But if a increases enough and E shifts enough to leave our starting point somewhere in R, below ~=(a’+ bL)/dL, then the increase in a may end up reducing employment. This final point can be checked by constructing a Lyapunov function for (II,). u(fi
,
v’)=
‘_d”,’
bL+a’P
-2
+gv2
(20)
is a Lyapunov function for system (II,) (A.7). Any level curve for U() forms an elipse with center at the new full employment equilibrium (p*‘, V*‘), principal axis at ~=(a + bL)/dL, and minor axis V= kdL. All paths in R, crossing any U() = U, pass from the outside to the inside of the level curve UO. Consider fig. 7. All paths which originate on the level curve U() = U,-, must
I II
a'tbi dC ’
Fig. 7
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terminate somewhere on the segment running from (p*‘, V*‘) to b. Paths which originate on U, between c and d must end up on E with V and L above their initial levels. But paths which originate between e and c may follow a path like that emanating from f; where V and L end up lower at the new equilibrium, h. The implication is this: If the system is at a less than full employment equilibrium, a series of ‘small’ increases in M or government spending (a) will lead to a full employment equilibrium (A.8), but one or more big jumps in a might actually reduce employment. On first reflection, this may seem counterintuitive, but it is an example of a common phenomenon in nature. One can’t, for example, till a teaspoon with a firehouse. But why do we get this result in the present case? The only complete answer is (II) itself, but on a common sense level, the reason might be this: A substantial increase in a raises demand enough to draw inventories down from their original equilibrium level. This, plus the increase in S’, causes prices to rise more than they would if production could keep pace with the higher sales and inventory demand. In the end, prices rise so much that real demand is below its original level. Arguments for gradualism in countercyclical policy are commonplace, but theoretical support of the sort given here is unique. 4. Conclusion
We have constructed a dynamic macro model with inventories, flexible prices, rational sales expectations, and trading out of equilibrium. On a technical level, the model has several unique features: There are multiple switching lines which don’t intersect at an equilibrium, the regimes or regions are not connected, and the equilibrium is a set, bordering on a switching line and otherwise contained in a single region. Of more direct economic interest, it is found that less than full employment equilibria are possible, even highly likely, in the intermediate term, i.e., while prices are flexible and nominal wages are fixed. If we reduce wages parametricly, starting from a less than full employment equilibrium, employment will drop - even though on a partial equilibrium level ALd/AwsO. Finally, it is shown that increasing aggregate demand will bring us from a less than full employment equilibrium to full employment, if the increase is (in a well defined sense) gradual. It is obvious that there is room for all sorts of continued research and progress in this area. It would be easy to say that we should include bonds, or speculative inventory demand, or simultaneously flexible wages and prices; but given the complexity of the present model, it is hard (for me) to imagine how any part of this structure could be made significantly more complex without, in the interest of manageability, making some other part less complex.
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Appendix
(A.].) The loci $‘i intersects & on 7: Proof: Substitute (13) into (14) to find that V= kdL at the point of intersection of tt and Iji. Then from (11) we see that at p = (bL + a)/dL, I/ = kdL on 7: also. (A.2.) @i cuts T from below at (p*, V*). Prooj: The slope of 6: at (p*, V*) is - k(dL)‘/(a+ bL), while that of T is -( 1 + k)(dL)‘/(a+ bL). Hence, fiz is flatter than T at (p*, V*). (A.3). In isolation, the system (II,) has a unique, locally stable equilibrium at (p*, I’*). Proof. Uniqueness is obvious. Defining ~5and P to be deviations from p* and I’*, (II,) is locally approximated by
(IL)
c)(1,
E,lj;:j. bL+a
It is obvious from inspection determinant.
that (IIJ
has a negative trace and positive
(A.4.) P!(&) intersects T at exactly two points. Proof: Set (11) equal to (16) and rearrange terms to get: p2Ja+(2+k)bL] dt
+ (l+k)b(a+bL)=O d2L
This factors to
yielding the roots given in the text. (k-7.) Proof
-(l
If (a+ bL)/dL>(l + k)b/d, then @(&!) is flatter at (p*, V*) than T is At p=(a+ bL)/dL, the slope of v: is - kd2L2/a, and the slope of T is +k)d’L’/(a+bL). If (l+k)b
’
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and cancelling terms, we have
kbL c a.
So, adding ka to each side, we have -(l+k)a< Then rearranging -
-k(a+bL).
terms and multiplying
(1 + k)d2L2 (a+bL)
<
both sides by d2L2, we have
- kd2L2 a ’
hence @ is flatter than T where they intersect at p =(a + bL)/dL. argument shows that the opposite is true at p=( 1 + k)b/d.
A similar
Ld on E is Ld = V/dk. Proof: Using (16) to get p = (adk + bV)/d y we substitute this expression for p into (10) to get Ld= V/dk. (A.6.)
(A.7.)
The function (20)
is a Lyapunov function. Proof: U() is clearly continuous and using (II& from (A.3), we find dU dt=
-2k(dQ4 (bL+a’)’
and positive definite,
so U is decreasing along any path in R,. (A.8.) A series of ‘small’ increases in a leads to full employment. Proof Let the system begin with a=a,,p=po, and V = V, = ka,d/(dp, - b) < kdL. This places the system at a less than full employment equilibrium on E,= with L= L, = VJdk. Now increase a to a,, with {(P, 4 1V=hs-W~-b)l~ a,ca, =(kdZ/VJa,. Since Vo< kdE, we know that there exists such an a,. In fact, this will be true at any t with t;c kdL or L, CL. The new a, defines a new E, and a new p:=(al + bL)/dL; and the value chosen for a, guarantees that po=p:, so (po, V,) is in the new region R, - in the sector which has V and L increasing along any path. Now let the system move to E,, where L= L, >L,. If we carry on in this fashion, increasing a, as long as a suitable a1+1 can be found, we get a monotone increasing sequence (L,). Since (L,) is
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bounded above by 1, the sequence must have a limit, say L+. But L+ must equal L, for if L+ CL, our rule for changing a indicates that there is a higher a leading to L’> L+, and the process is not yet complete. References Amihud, Y. and H. Mendelson, 1982, The output-inflation relationship, Journal of Monetary Economics 9, 163-184. Amihud, Y. and H. Mendelson, 1983, Price smoothing and inventory, Review of Economic Studies L, 87-98. Blinder, A., 1977, A difficulty with keynesian models of aggregate demand, in: A. Blinder and P. Friedman, eds., Natural resources, uncertainty and general equilibrium systems: Essays in honor of Rafael Lusky. Blinder, A., 1980, Inventories and the Keynesian macro model, Kyklos 33, 585-614. Blinder, A., 1982, Inventories and sticky prices: More on the microfoundations of macroeconomics, American Economic Review 72, 334-348. Blinder, A. and S. Fischer, 1981, Inventories, rational expectations and the business cycle, Journal of Monetary Economics 8, 277-304. Brunner, K., A. Cukierman and A. Meltzer, 1980, Money and economic activity, inventories and business cycles, Mimeo. (Graduate School of Management, University of Rochester, Rochester, NY). Chaudbury, A., 1979, Output, employment, and inventories under general excess supply, Journal of Monetary Economics 5,505-514. Eckalbar, J., 1980, The stability of non-Walrasian processes: Two examples, Econometrica 48, 371-386. Eckalbar, J., 1981, Stable quantities in lixed price disequilibrium, Journal of Economic Theory 25, 302-3 13. Eckalbar, J., 1985, Inventory fluctuations in a disequilibrium macro model, Economic Journal, forthcoming. Flugge-Lotz, I., 1968, Discontinuous and optimal control (McGraw-Hill, New York). Friedman, B., 1979, Optimal expectations and the extreme information assumptions of ‘rational expectations’ macro models, Journal of Monetary Economics 5, 2341. Green, J. and J.J. Laffont, 1981, Disequilibrium dynamics with inventories and anticipatory price-setting, European Economic Review 16, 199-221. Honkapohja, S. and T. Ito, 1980, Inventory dynamics in a simple disequilibrium macro model, Scandinavian Journal of Economics 82, 184-198. Honkapohja, S. and T. Ito, 1983, Stability with regime switching, Journal of Economic Theory 29, 22-48. Keynes, J., 1936, The general theory of employment, interest, and money (Harcourt, Brace and World, New York). Lovel, M., 1961, Manufacturer’s inventories, sales expectations, and the acceleration principal, Econometrica 29,293-314. Metzler, L., 1941, The nature and stability of inventory cycles, Review of Economics and Statistics 23, 113-129. Reagan, P., 1982, Inventory and price behavior, Review of Economic Studies XLIX, 137-142. Siminovits, A., 1982, Buffer stocks and naive expectations in a non-Walrasian dynamic macromodel: Stability, cyclicity and chaos, Scandinavian Journal of Economics.
INVENTORIES
IN A DYNAMIC MACRO FLEXIBLE PRICES*
MODEL
WITH
John C. ECKALBAR California
State University,
Chico,
California
95926,
CA
Received June 1984, tinal version received November 1984 This paper presents a disequilibrium macro model with inventories, rational expectations, and flexible prices. It is found that less than full employment equilibria are possible, even highly likely, in the intermediate term and that wage reductions will cause employment to drop. Increasing aggregate demand will raise employment only if it is done gradually. On a technical level, we have multiple switching lines, disconnected regimes, and a connected equilibrium set which borders on a switching ljne and otherwise lies in a single region.
1. Introduction Just a few years ago, there were hardly any theoretical macro models with inventories.’ In retrospect this is hard to understand in view of the fact that economists, at least since Keynes, ’ have felt that inventories play a key role in macro dynamics, and statistical studies have consistently shown that inventory fluctuations account for a substantial fraction of cyclical output variations. Now, thanks largely to a series of papers by Alan Blinder, we are seeing a burst of interest in inventory theoretic macro modeling. Virtually every major segment of macro theory has been influenced - there are IS-LM based models [Blinder (1977,1980)], rational expectations models [Blinder and Fischer (1981), Green and Laffont (1981), Brunner, Cukierman and Meltzer (1980)], and disequilibrium models [Honkapohja and Ito (1980), Eckalbar (1985), Simonovits (1982)].3 If there is a common weakness to these models as a group, it is their treatment of prices. Existing models seem to either treat prices as fixed, sometimes ignoring them altogether [Honkapohja and Ito (1980), Eckalbar 71985), Simonovits (1982), Chaudbury (1979), Blinder (1977)], or they assume that prices adjust instantly to equate demand and *Research reported here was partly funded by a grant from the National Science Foundation. ‘Exceptions: Metzler (1941), Love1 (1961). ‘The General Theory of Employment, Interest and Money (1936, p. 318). % addition, there is considerable work under way on the microeconomics of inventories: Reagan (1982), Amihud and Mendelson (1982, 1983). 00142921/85/%3.30 0 1985, Elsevier Science Publishers B.V. (North-Holland)
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supply, or the expected values of demand and supply [Blinder and Fischer (1981), Brunner, Cukierman and Meltzer (1980), Green and Laffont (1981)]. The present paper incorporates flexible prices, without resorting to the unrealistic assumption of continuous market clearing. The firm is assumed to be a price setter. It has a fixed target inventory-expected sales ratio; and it raises (lowers) price whenever actual inventories are below (above) their target level. It is found that in the ‘intermediate term’, i.e., while nominal wages are fixed, less than full employment equilibria are possible, even though real wages are flexible. In the long run, (i) wage reductions are likely to lower the employment level; and (ii) a ‘moderate’ program of demand expansion can lead to full employment, while an ‘excessive’ boost in demand can be counterproductive.4 On the technical side, the model presented here has several novel features. (i) There are multiple switching lines, which do not intersect at an equilibrium. (ii) The equilibrium is a connected set. (iii) There are branching points, or points of indecision in the phase space. (iv) Individual regime domains are disconnected. Section 2 gives the model. Section 3 outlines the comparative statics, and section 4 is the conclusion. 2. The model
The model contains two goods, labor and output, and two sectors, households and firms. The household offers a fixed quantity of labor, L, each period and buys output according to the sales equation, a+cwL
s=p,
(1)
where a and c are constants (a>O; 0
CM, + CWL P
’
and ‘excessive’ are given precise definitions in the text,
(2)
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where M, is beginning period money holdings. If the household always receives dividends equal to profits, M, is constant.’ Then, since we don’t have any pretense to a monetary theory here, we will set CM, = a to yield (1). The firm is responsible for hiring labor, producing output, making sales, and adjusting prices. We assume that the firm forms expectations of upcoming sales and, if it is profitable to do so, tries to hire enough labor to provide for these sales and leave itself with inventory, K equal to kS’, where k is a positive constant and Se is the expected level of sales. Of all the simple assumptions one could make about target inventory levels, the fixed target ratio to sales seems most defensible. The actual ratio of inventories to sales is trendless over the past forty years, and cyclical turns in the ratio can be accounted for by slowly adjusting expectations.6 For simplicity, we assume that output, Q, is given by Q=dL,
d>O.
(3)
If w/p>d, production is unprofitable, and labor demand will be zero. But if w/p 5 d, the firm will wayt to produce Q=max[(l+k)S’-KO].
(4)
Two remarks: (i) (1 + k)Se- V is the output level which would provide for all anticipated sales and bring inventory to its target level, kS’. We are implicitly assuming that the firm makes an effort to adjust inventory completely within the period. Feldstein and Auerbach offer empirical support for this assumption. (ii) If V is high relative to S”, negative output would be called for were it not for the max condition in (4). The above considerations give rise to a labor demand equation of the following form: Ld=min($-d,O)*[l/f-d]max[(l+k)S’-KO]*-$ Note that high inventory levels tend to reduce desired output and derived labor demand, as most micro theoretic discussions have shown, and note that d Ld/d (w/p) s 0, with other variables constant. As is usual in this class of models, we assume that the quantity of labor actually hired is the lesser of the quantities supplied and demanded, Thus: L = min (L, Ld),
and the firm may be unable to produce everything it wishes. ‘See my ‘Stable Quantities . . .’ for an alternative formulation. %ee Eckalbar (1985).
(6)
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The dynamics of the model are as follows: Wages, prices, inventories, and sales expectations give rise to Ld, then min(Ld, L) gives L, which determines S and Q. The values for S and Q imply the change in inventories, Q-S. The firm is assumed to be a price setter. We will suppose that the firm compares desired inventories, kS”, to actual inventories, r! and adjusts prices upward (downward) if kS’ is greater (less) than K Recent papers by Reagan (1982), Blinder (1982), and Amihud and Mendelson (1982,1983) offer a solid dynamic programming microfoundation for our price adjustment assumption. If we assume that wages adjust according to the excess demand in the labor market and that sales expectations change according to an as yet unspecified function f( ), we have the following dynamical system: ti=h(Ld-L), fi = g( kS” - I’), (1)
P = d. min (Ld, L) - [a + cw min (Ld, L)]/p,
where Ld is given by (5) and h() and g() are sign preserving. System (I) has three switching surfaces [Ld =L, w/p=d, and (1 + k)Se= P’l in four dimensions. A model of this complexity is, at least at the present, beyond the range of existing disequilibrium techniques.’ However, two commonly used assumptions are sufficient to render the system tractable while still preserving enough of the ‘detail to keep the problem interesting. First, we assume that w is fixed in the short run. Later, if we find that the system settles at a less than full employment equilibrium, we will consider parametric changes in w - but for the moment, w will be thought of as infinitely slow relative to the remaining variables. Second, we assume that sales expectations are formed ‘rationally’. In the present context this means that if the firm hires labor on the assumption that sales will equal Se, sales will in fact equal S,, when that quantity of labor is hired. In most works of this sort, ‘rationality’ is a matter of faith. But if a theorist cannot derive rationality from some defensible learning process, then the assumption seems unwarranted. Accordingly, we digress for a moment to consider learning. Following Benjamin Friedman, we take adaptive expectations to be an approximation of optimal least-squares learning. Imagine that with p and V fixed, households and firms go through tatonnement with respect to S and Ld. With S’ adjusting adaptively, we have the following system for the region ‘See Honkapohja and Ito (1983), Eckalbar (1980).
J.C.
where O
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CL.
S,e=SC-l+a(S,_,-S:-,),
W
in dynamic
L:+l+k)s:-V-J,
O
a+CW,L; P .
This yields the first order difference equation (7) which is convergent if p>cw,(l +k)/d. This restriction will be examined shortly. In the regions where Ld = 0 and L=L, the difference equation becomes SF= (1 - a)S:- I + constant, so no additional restriction is necessary here to guarantee convergence to ‘rationality’. This sort of no-trade tatonnement in S and Ld (‘tell me what you’re going to do and I’ll tell you what I’m going to do’) is not a completely satisfactory characterization of the learning process. As an alternative, we might suppose that prices and wages are fixed, but that trade actually takes place so that V and S’ both change during learning. Assuming continuous adjustment, we have the following expectation adjustment system:
(EJ
Se=a(S-Se),
v=Q-S.
In the region where 0 < Ld -CL, we have:
where p and P are deviations from equilibria, and H is a column vector of constants. The Routh-Hurwitz conditions show that the learning process in (E;) is convergent if p> [cw,( 1 + k) -cwe]/( 1 +a)d. As a varies from 0 to 1, the lower limit for p varies from cw,/d to cw,(l + Q/d. A comment on this shortly. For the regions L=O and L= E, no additional restrictions are required to yield a convergent learning process.
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Notice that convergence to ‘rational’ expectations is problematic only in the region where 0 < Ld
ka + kb min (Ld, L) _ I/ P
,
where for simplicity we have defined b =cwo, and we will now take g to be a positive constant. Our method will be to investigate this system one regime at a time and then patch the various regimes together for a look at the whole system. First, consider R,={(p, v)) w,/d>p}. In RI the real wage exceeds the marginal product of labor, and profitable production is impossible. It follows from (5) that Ld=O in RI; thus in this subspace (II) becomes (III -3) V= -$
;-Jf.
Ij=g
,
(
>
This gives rise to the flow depicted’in fig. 1. Note that V is falling everywhere in R,, and that p is falling (rising) everywhere above (below) the locus A, = {(p, v) 1V= ka/p}. The latter fact rules out negative prices. If p> we/d, then by (5) we have Ld=fmax[(l If O
+k)S-
KO].
(8)
we have
1 .
Solving for Ld using rational expectations, we have L,js(l+k)a-PV dp-(1 +k)b’
Note that (10) is discontinuous
O
at p = (1 + k)b/d.
(9)
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d
Fig. 1
Defining T = {(p, the equation
V) 1 Ld = E},
we find that the switching line T is given by
The locus T is asymptotic to the vertical axis and to the line V= -dL. Checking (10) we see that (where it exists)
aP
-P
z=
dp-(1 +k)b’
(12)
hence a small increase in V from a point on T will yield Ld
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r
K
d \
T
Fig. 2
p=( 1 + k)b/d or below T and left of p=( 1 +k)b/d. Given (12), we know that Ld
J.C. Eckalbar,
014)
v=
(j-b (
jQ! P>
P’
in dynamic
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model
+g[~(a+bwj.
System(I1,) defines two loci, pi= {(p, v) ERq/ i’=O} Points in $‘i are on the vertical line given by
and @z= {(p, v) ER,l/j=O}.
bZ+a P=dL.
Everywhere in R, to the left (right) of this line inventories are falling (rising). The locus i/z may fall on either side of the line p =(l + k)b/d. In order to keep our figure drawing to a minimum, we will assume that (bL+ a)/dL> (1 + k)b/d and leave it as an exercise for the reader to redraw the figures for the other case. The locus @i is a rectangular hyperbola with both axes as asymptotes, it is given by the line
V=k(a+bf) P
(14)
’
The loci J$ and vz meet on T (A.l)* at the full employment
equilibrium
(15)
.
The locus iz cuts T from below, passing through R, into R, as shown in fig. 3 (A.2). (Here and elsewhere, a locus will be shown as a dashed line over any portion of itself outside its domain.) It is clear from inspection of (II,) that p is falling everywhere in R, above pz, and vice versa. This produces the vector field shown in fig. 3. If the full employment equilibrium were in the interior of Rq, it would be locally asymptotically stable (A.3). But we know from (A.l) that (p*, I’*) falls on the border between R, and R,, so any stability check is premature at this point. The only region remaining is R2, within which 0 -CL* = L < L. Substituting (10) for L* in (II) we have
1-a
p’
‘To preserve the flow of the text, numerous minor assertions will be proven in the appendix and marked (A.l), (A.2), etc.
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-! P: V’) ---_ R2
A+
\
Fig.
Defining ti given by v=
and fig analogous with vi and rji, we find that both loci are kad dp-b’
(16)
The implication is that any point in R, satisfying (16) is a less than full employment equilibrium. But does I’!#) run through R,? Yes. k’r&$) is a rectangular hyperbole with asymptotes at p= b/d and V=O. It intersects T at exactly two points - (p*, V*) and (b(1 + k)/d, ad/b) (A.4). At (p*, V*) the locus i/8(@8) is flatter than T; while at (b(1 + k)/d, ad/b), it is steeper than T (A.5). This implies that @(pi) passes into one section of R, as shown in fig. 4. The significance of this is that the complete system, (II), does not have a unique equilibrium - it has an equilibrium locus, with a full employment equilibrium as an endpoint and an infinity of less than full employment equilibria trailing downward to the right. For convenience, we define the equilibrium set E = {(p, V)l(p, V) ER,, V = kad/(dp - b)}.
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a+bi dt Fig. 4
We are now in a position to complete the dynamic picture of (II) by in the flow in R,. From (II,), we see that
aif
-(dp-b)
%=dp-(1
+k)b’
(17)
Since p > b/d in R,, (17) is positive if p < (1 + k)b/d and negative otherwise.g ‘Note the frequency that the term dp-( 1+ k)b has come up in the text. If p>( I+ k)b/d, optimal OLS learning will always converge, p=( 1+ k)b/d constitutes a regime border, etc. In Honkapohja and Ito (1980) and Simonovitz (1982), where prices are suppressed, an equivalent term (a-b(1 +p)) proves to be critical, but they are forced to simply assume that it takes one sign or another. [See Honkapohja-Ito (1980, p. 189).] Having prices variable is clearly superior in that: (i) We don’t have to restrict ourselves to one sign for dp-( 1 +k)b; as it changes sign, we simply switch regions. (ii) We are able to shed some light on why dp-( 1 + k)b is critical, i.e., in terms of learning and regime switching.
E.E.R.-
D
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This means that in R, near E, V is decreasing increasing if it is below. See fig. 4. A similar argument would show that
model
if it is above @ and
ati -d&-b)
W
a=+(l+k)b
which gives the flow direction for p in fig. 4. Figs. 14 can be combined to give In the interest of holding down the dispense with the variety of switching of the flow. This is done in fig. 5.
R, on either side of t!&ji).
Again, see
a complete picture of the flow of (II). complexity of the resulting figure, we loci and show only the general nature
Three technical notes: First, on the line p=( 1+ k)b/d, between A, and fii, the system has ‘branching points’ or ‘points of indecision’.” That is, points slightly right of VI
I
P
0
Fig. 5 ‘OSee Flugge-Lotz.
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this segment move down and right (into R4), ‘while points slightly left move down and left (into R3). The problem is trivial, since no path moves onto this segment, and any displacement from the segment leads to a fully determinant path. Second, nothing in the analysis precludes inventories from becoming negative. This might be dealt with in two ways: (i) we might impose a border restriction on t such as i/z0 for T/=0; or (ii) we might let V become negative and think of V as simply a backlog and S as an order rather than a sale. The latter is clearly a better solution, but can that upset the stability of E? No. if we cross below V=O, it will have to be at some p <(a +bE)/dL, but then p will always be increasing along such a path, so we must sooner or later find p > (a+ bE)/dL. If so, V will begin increasing and will sooner or later cross V=O again. Hence, V is bounded below along any path. Prices could never become negative, since 1ir1+~ S= 03, and price change is positively linked to S. See (II, -3). Third, the regions where labor demand and output are zero (R, and R3) are included for analytical,completeness. They are largely a byproduct of the assumption of linear technology, and they are of little economic relevance. Note that these regions play no role in the local dynamics around E, which is contained in R, and R,. Finally, the stability of E can be rigorously established with a proof which follows the method of theorem 5 of my Econometrica paper, but in this case the proof will be dispensed with, since stability of E is obvious from visual inspection. 3. The comparative
statics
(i) We have been supposing that w is fixed as the system makes its way to E, i.e., that wages are infinitely slow. But what if we settle at some unemployment equilibrium, and then the wage falls? Does the wage adjustment move the system closer to full employment? Until now the wage has been subsumed into b via b=cw,. So a reduction in the wage translates into a drop in b. Looking back at the figures, we see that a reduction in b slides the equilibrium locus, along with every other reference line, to the left. So if we began at an equilibrium like point x in fig. 6, a one time drop in w would lead us to a point on E’ somewhere between y and z. Notice that V will certainly be lower at the new equilibrium. This is significant. It can be shown (A.6) that on E, employment and labor demand are given by L = Ld = V/kd.
Thus, equilibrium
employment
(1%
is, a fixed multiple
of the equilibrium
inven-
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V
V’kdi
c1
P
Fig. 6
tory level. It follows from this that employment is less along the segment yz than at x, or that the wage reduction makes matters worse in the labor market. The reason for this is that when wages drop, the household’s purchasing power also drops; and the firm’s labor demand is predicated upon sales to households. So as households buy less, firms hire fewer workers. (ii) Thus far we have not built in a government sector, but it is easy to see how that might be done. We have already shown that autonomous sales, a, might be taken to be cMo, thus an increase in the supply of money would transalte into an increase .in a. Similarly, since a is simply nominal autonomous sales demand, an increase in government demand would be equivalent to an increase in a. When a increases, most of the loci, including E, shift to the right. If such a policy is undertaken when the system is initially at a less than full employment equilibrium, then after the shift in E, we would be at an ‘initial’ point somewhere in R, or R4 below V= kdz. We can be sure that an increase in u will cause prices to rise, but the employment outcome is less certain. If the shift in E leaves our initial point
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somewhere in R, or in R, above P=(a’+bt)/dL (where a’ is the new higher level of a), then I/ will be everywhere increasing along the path to E’, and we must end up with V and, therefore, L above their original values. But if a increases enough and E shifts enough to leave our starting point somewhere in R, below ~=(a’+ bL)/dL, then the increase in a may end up reducing employment. This final point can be checked by constructing a Lyapunov function for (II,). u(fi
,
v’)=
‘_d”,’
bL+a’P
-2
+gv2
(20)
is a Lyapunov function for system (II,) (A.7). Any level curve for U() forms an elipse with center at the new full employment equilibrium (p*‘, V*‘), principal axis at ~=(a + bL)/dL, and minor axis V= kdL. All paths in R, crossing any U() = U, pass from the outside to the inside of the level curve UO. Consider fig. 7. All paths which originate on the level curve U() = U,-, must
I II
a'tbi dC ’
Fig. 7
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terminate somewhere on the segment running from (p*‘, V*‘) to b. Paths which originate on U, between c and d must end up on E with V and L above their initial levels. But paths which originate between e and c may follow a path like that emanating from f; where V and L end up lower at the new equilibrium, h. The implication is this: If the system is at a less than full employment equilibrium, a series of ‘small’ increases in M or government spending (a) will lead to a full employment equilibrium (A.8), but one or more big jumps in a might actually reduce employment. On first reflection, this may seem counterintuitive, but it is an example of a common phenomenon in nature. One can’t, for example, till a teaspoon with a firehouse. But why do we get this result in the present case? The only complete answer is (II) itself, but on a common sense level, the reason might be this: A substantial increase in a raises demand enough to draw inventories down from their original equilibrium level. This, plus the increase in S’, causes prices to rise more than they would if production could keep pace with the higher sales and inventory demand. In the end, prices rise so much that real demand is below its original level. Arguments for gradualism in countercyclical policy are commonplace, but theoretical support of the sort given here is unique. 4. Conclusion
We have constructed a dynamic macro model with inventories, flexible prices, rational sales expectations, and trading out of equilibrium. On a technical level, the model has several unique features: There are multiple switching lines which don’t intersect at an equilibrium, the regimes or regions are not connected, and the equilibrium is a set, bordering on a switching line and otherwise contained in a single region. Of more direct economic interest, it is found that less than full employment equilibria are possible, even highly likely, in the intermediate term, i.e., while prices are flexible and nominal wages are fixed. If we reduce wages parametricly, starting from a less than full employment equilibrium, employment will drop - even though on a partial equilibrium level ALd/AwsO. Finally, it is shown that increasing aggregate demand will bring us from a less than full employment equilibrium to full employment, if the increase is (in a well defined sense) gradual. It is obvious that there is room for all sorts of continued research and progress in this area. It would be easy to say that we should include bonds, or speculative inventory demand, or simultaneously flexible wages and prices; but given the complexity of the present model, it is hard (for me) to imagine how any part of this structure could be made significantly more complex without, in the interest of manageability, making some other part less complex.
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Appendix
(A.].) The loci $‘i intersects & on 7: Proof: Substitute (13) into (14) to find that V= kdL at the point of intersection of tt and Iji. Then from (11) we see that at p = (bL + a)/dL, I/ = kdL on 7: also. (A.2.) @i cuts T from below at (p*, V*). Prooj: The slope of 6: at (p*, V*) is - k(dL)‘/(a+ bL), while that of T is -( 1 + k)(dL)‘/(a+ bL). Hence, fiz is flatter than T at (p*, V*). (A.3). In isolation, the system (II,) has a unique, locally stable equilibrium at (p*, I’*). Proof. Uniqueness is obvious. Defining ~5and P to be deviations from p* and I’*, (II,) is locally approximated by
(IL)
c)(1,
E,lj;:j. bL+a
It is obvious from inspection determinant.
that (IIJ
has a negative trace and positive
(A.4.) P!(&) intersects T at exactly two points. Proof: Set (11) equal to (16) and rearrange terms to get: p2Ja+(2+k)bL] dt
+ (l+k)b(a+bL)=O d2L
This factors to
yielding the roots given in the text. (k-7.) Proof
-(l
If (a+ bL)/dL>(l + k)b/d, then @(&!) is flatter at (p*, V*) than T is At p=(a+ bL)/dL, the slope of v: is - kd2L2/a, and the slope of T is +k)d’L’/(a+bL). If (l+k)b
’
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then cross multiplying
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and cancelling terms, we have
kbL c a.
So, adding ka to each side, we have -(l+k)a< Then rearranging -
-k(a+bL).
terms and multiplying
(1 + k)d2L2 (a+bL)
<
both sides by d2L2, we have
- kd2L2 a ’
hence @ is flatter than T where they intersect at p =(a + bL)/dL. argument shows that the opposite is true at p=( 1 + k)b/d.
A similar
Ld on E is Ld = V/dk. Proof: Using (16) to get p = (adk + bV)/d y we substitute this expression for p into (10) to get Ld= V/dk. (A.6.)
(A.7.)
The function (20)
is a Lyapunov function. Proof: U() is clearly continuous and using (II& from (A.3), we find dU dt=
-2k(dQ4 (bL+a’)’
and positive definite,
so U is decreasing along any path in R,. (A.8.) A series of ‘small’ increases in a leads to full employment. Proof Let the system begin with a=a,,p=po, and V = V, = ka,d/(dp, - b) < kdL. This places the system at a less than full employment equilibrium on E,= with L= L, = VJdk. Now increase a to a,, with {(P, 4 1V=hs-W~-b)l~ a,ca, =(kdZ/VJa,. Since Vo< kdE, we know that there exists such an a,. In fact, this will be true at any t with t;c kdL or L, CL. The new a, defines a new E, and a new p:=(al + bL)/dL; and the value chosen for a, guarantees that po=p:, so (po, V,) is in the new region R, - in the sector which has V and L increasing along any path. Now let the system move to E,, where L= L, >L,. If we carry on in this fashion, increasing a, as long as a suitable a1+1 can be found, we get a monotone increasing sequence (L,). Since (L,) is
J.C. Eckalbar,
Inventories in dynamic
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bounded above by 1, the sequence must have a limit, say L+. But L+ must equal L, for if L+ CL, our rule for changing a indicates that there is a higher a leading to L’> L+, and the process is not yet complete. References Amihud, Y. and H. Mendelson, 1982, The output-inflation relationship, Journal of Monetary Economics 9, 163-184. Amihud, Y. and H. Mendelson, 1983, Price smoothing and inventory, Review of Economic Studies L, 87-98. Blinder, A., 1977, A difficulty with keynesian models of aggregate demand, in: A. Blinder and P. Friedman, eds., Natural resources, uncertainty and general equilibrium systems: Essays in honor of Rafael Lusky. Blinder, A., 1980, Inventories and the Keynesian macro model, Kyklos 33, 585-614. Blinder, A., 1982, Inventories and sticky prices: More on the microfoundations of macroeconomics, American Economic Review 72, 334-348. Blinder, A. and S. Fischer, 1981, Inventories, rational expectations and the business cycle, Journal of Monetary Economics 8, 277-304. Brunner, K., A. Cukierman and A. Meltzer, 1980, Money and economic activity, inventories and business cycles, Mimeo. (Graduate School of Management, University of Rochester, Rochester, NY). Chaudbury, A., 1979, Output, employment, and inventories under general excess supply, Journal of Monetary Economics 5,505-514. Eckalbar, J., 1980, The stability of non-Walrasian processes: Two examples, Econometrica 48, 371-386. Eckalbar, J., 1981, Stable quantities in lixed price disequilibrium, Journal of Economic Theory 25, 302-3 13. Eckalbar, J., 1985, Inventory fluctuations in a disequilibrium macro model, Economic Journal, forthcoming. Flugge-Lotz, I., 1968, Discontinuous and optimal control (McGraw-Hill, New York). Friedman, B., 1979, Optimal expectations and the extreme information assumptions of ‘rational expectations’ macro models, Journal of Monetary Economics 5, 2341. Green, J. and J.J. Laffont, 1981, Disequilibrium dynamics with inventories and anticipatory price-setting, European Economic Review 16, 199-221. Honkapohja, S. and T. Ito, 1980, Inventory dynamics in a simple disequilibrium macro model, Scandinavian Journal of Economics 82, 184-198. Honkapohja, S. and T. Ito, 1983, Stability with regime switching, Journal of Economic Theory 29, 22-48. Keynes, J., 1936, The general theory of employment, interest, and money (Harcourt, Brace and World, New York). Lovel, M., 1961, Manufacturer’s inventories, sales expectations, and the acceleration principal, Econometrica 29,293-314. Metzler, L., 1941, The nature and stability of inventory cycles, Review of Economics and Statistics 23, 113-129. Reagan, P., 1982, Inventory and price behavior, Review of Economic Studies XLIX, 137-142. Siminovits, A., 1982, Buffer stocks and naive expectations in a non-Walrasian dynamic macromodel: Stability, cyclicity and chaos, Scandinavian Journal of Economics.