Endogenous Oscillations in a Discrete Dynamic Model with Inventory

Introducing the producer's intertemporal optimizing behavior, we extend the Eckalbar Disequilibrium Macro-Model (1985) and reconsider the dynamic features of the modified model. We concern ourselves with the existence of inventory cycles when the expectations are formed adaptively. The endogenous inventory cycle is detected using the Hopf bifurcation theorem in which a bifurcation parameter is an adaptive coefficient. It is also demonstrated that the generated cycle is subcritical.


INTRODUCTION
This study analyzes non-linear dynamics of a simple disequilibrium macro-model with inven- tories.The main purpose is to investigate what role the profit-maximizing firm plays to generate cyclic inventory dynamics.The model has linear demand and non-linear supply, the latter of which is an outcome of intertemporal profit max- imization by the firm.The Hopf bifurcation the- orem is used to demonstrate the existence of endogenous inventory cycles.Further, an exam- ple is presented to show that the generated cycles are subcritical when the production cost function is linear.
Inventory-theoretic macro-models are developed in the framework of disequilibrium econom- ics.Several stability or unstability results have been established.This study extends Eckalbar   (1985) by introducing the optimal behavior of the profit maximizing firm.Eckalbar constructs a continuous-time macro-model in which expectations on sales are adaptively adjusted and estab- lishes the existence of limit cycles, applying the Poincar6-Bendixson theorem.The dynamic sys- tem employed is non-linear, but sources of non- linearity are exogenously determined.That is, lower and upper bounds of variables such as full- employment output and non-negative employ- ment are exogenously introduced and work to Tel.: + 81/(0)25-262-6551.Fax: + 81/(0) 25-263-3262.E-mail: eakio@hle.niigata-u.ac.jp.See Honkapohja and Ito (1980), Simonovits (1982), Eckalbar (1985) and Franke and Lux (1993).203 prevent the unstable behavior from expanding dynamics.Little is known about a source of such globally.In particular, these exogenous variables non-linearity.To go one step further, we con- define switching lines to divide the phase space struct a micro-foundation of the Eckalbar into subregions and make the system a sort of macro-model and shed light on its non-linear dynamical hybrid.Thus, in one region divided by structure in which the optimal behavior of the the switching lines, one unstable subsystem gov- profit maximizing firm plays an important role erns the dynamic variables and drives these away for generations of cyclic dynamics.
from the equilibrium point.In another region, The fundamental characteristics of our model another stable subsystem governs the same vari- are similar to those of the Zhang model as well ables and drives these back to the region in as the Poston model and thus those of the which the equilibrium point exists.The dynamic Eckalbar model.But there are many deviations variables oscillate back and forth in these re- from these models.First, our model is cast in gions.When the stabilizing force is balanced discrete time, whereas their models are in contin- against the unstabilizing force, the cyclic dy- uous time.It is worthwhile to consider a discrete namics can emerge in such models.Coexistence version of the Eckalbar model because the dyof opposite-directed dynamic forces is due to the namics generated by a discrete-time system is exogenous factors,  significantly different from the dynamics by a There are some directions in which the Eckalbar continuous-time system.2Second, we derive linear model is extended.In the first half of their choice-theoretically the producer's behavioral study, Poston et al. (1992) refine on the Eckalbar functions based on the intertemporal profit max- model and clarify the conditions for sustained imization.This provides a microeconomic foun- oscillation.Zhang (1989) and Poston et al. (1992)  dation of the supply side of the model.In the (in the latter half) generalize the Eckalbar's piece- models of Zhang and Poston et al., the non-lin- wise linear model with the purpose of elucidating earity of the desired stock adjustment function or endogenous oscillations independent of the exo- the inventory investment function is, as men- genous factors.Zhang introduces a non-linear tioned above, assumed directly on the producer's adjustment function of the desired inventory behavior.Third, although we also apply a dis- while Poston et al. introduce a non-linear inven- crete-time Hopf bifurcation theorem to show en- tory investment function.In doing so, they re- dogenous oscillations, a bifurcation parameter in place the exogenously determined switching our model is a coefficient of the adaptive expecdynamic system of the Eckalbar model with the tations on sales, whereas it is the marginal pro- endogenously determined non-linear dynamic pensity to consume in the Zhang model.In this systems and demonstrate the existences of endo- study, it is demonstrated that the endogenous genous inventory cycles, the existences of which a inventory cycle can emerge when a set of adap-Hopf bifurcation theorem is applied to establish, tive coefficient and inventory-expectation ratio In those studies, however, non-linearity, which is crosses critical values for which the characteristic sufficiently strong to bring out cyclic dynamics, roots of the dynamical system become complex.has been assumed more or less directly on the The paper is organized as follows.Section 2 dynamical system or on the economic behavior constructs a basic model based on individual's of a particular body of agents.It has been optimizing behavior.Section 3 considers inventory known that the dynamic model endowed with a dynamics with adaptive expectations.Section 4 sufficient non-linearity can generate complex makes concluding remarks.

THE BASIC MODEL
This section recapitulates the fundamental struc- ture of Eckalbar's model and introduces the producer's dynamic optimizing behavior.A model has two traders and three commodities.Three commodities are aggregate consumption goods, labor, and money.The goods are assumed to be storable.Two trades are a consumer and a pro- ducer.3 Exchange takes place through money so that there are two markets: the consumption goods market and the labor market.The price of the consumption goods, p, and the wage rate, w, are exogenously fixed.These prices not being equilibrium one, demand is not necessarily equal to supply in each market.
For the sake of simplicity, we make two assumptions to determine the actual quantity traded in each market.First, the markets are assumed to operate sequentially so that the tra- ders enter the labor market and then the goods market.Second, actual transaction in a disequili- brium market is assumed to be determined by the minimum of supply and demand (i.e., the "min-rule" or "short-side" rule of disequilibrium theory).Accordingly, the traders find a difference between what they expect to trade at the start of a period and what they actually realize at the end.The difference not only determines the ini- tial level of buffer stocks in the following period but also affects the revision of expectations of the market state on which the traders base their economic decisions.Consequently, the macro- dynamics evolves.
We divide the remaining part of this section into three parts.In the first part, we describe the consumer's behavior.Since our emphasis of this study is placed on how the producer's optimal behavior affects the macro-dynamics, we specify the consumer's behavior as simple as possible.In the second, we describe the intertemporal optimal behavior of the firm.In the third, we examine determination of output, employment and inven- tory accumulation.

Consumer
We make a behavioral specification on the repre- sentative consumer in this subsection.For the sake of brevity, we do not formulate the utility maximization problem of the consumer but as- sume the inelastic supply of labor, N, 4 and the linear Keynesian-type expenditure function.Since the labor market operates first, the consu- mer knows if he is fully employed or not when he enters the goods market.Having the actual quantity traded in the labor market, L, the con- sumer makes a choice of consumption demand, S, by S(L) co + cL, (1) where Co > 0 is the demand for the goods in the case of unemployment and c is the marginal pro- pensity to demand with respect to employment.

Firm
We describe the producer's intertemporal profit maximization behavior in this subsection, y is the quantity of the consumption goods produced by using employed labor, L, with the conventional production function, y=F(L), h is an initial stock of inventory at the start of a decision period and h +1 the inventory carried-over to the following period.Hence the inventory accumula- tion equation is h+l =hd-y-S. (2) s is an expectation on sales that the producer forms before entering the goods market.We assume that We suppose that our economy is composed of a fixed number of identical producers and of identical consumers.Hence, the analysis focuses on the behavior of the representative producer and the representative consumer, each of whom is taken to reflect the corresponding aggregate behavior.
the producer has the desired level of inventory, denoted by/, to be a fixed ratio of expected sales to stock: /-3s, where fl > 0. (3) We make the fixed ratio assumption in order to clarify the firm's contributions to persistent cyclical behavior of inventory. 5As is seen later in the dynamic analysis, the firm's profit maximizing behavior leads to a non-linear dynamic system even under the fixed-ratio assumption.
The producer incurs two types of cost" the cost of producing output and the cost of holding inventory.We denote by C(y) the cost associated with producing y.Labor being the only input for the production function, it is the labor cost.That is, C(y) wF-l(y), where F-(y) is an inverse of the production function and denotes the quantity of labor necessary to produce output, y.We assume that the cost of holding inventory is asso- ciated with a deviation of an actual level of in- ventory from the desired level of inventory, and denote it by H(h-[0.V(h + 1) is the maximum of the expected profit that the producer achieves by employing the best policy from the next period and onwards. 6It accounts for the discount fac- tor.We make the following assumptions on these functions:
Assumption 1(1) states that the marginal pro- ductivity of labor is positive and further employ-ment brings about further but smaller production increases.As a result of this assumption, the marginal cost of production is positive and in- creasing.Assumption 1(2) states that as an actual level of inventories, h, deviates from the desired level of inventory, h, the cost of holding inven- tory increases due to the loss of the goodwill for negative deviation (i.e., h </) and due to the increase of the storage cost for positive deviation (i.e., h >/).Further, the convexity is assumed.
Assumption 1(3) describes that the imputed real values increases at a decreasing rate as inven- tories increase. 7  We solve the firm's optimization problem.The firm chooses production and inventory carried- over so as to maximize the expected profit, subject to the non-negative constrains on decision variables, y_>0 and h+l-h+y-s>_O, where the firm takes only intended change of inventory into account when choosing the opti- mal plan.The Lagrangian of the profit maximi- zation problem is d) + v + A{y max(0, s h)}. (6) Differentiating Eq. ( 6) with respect to y yields the first-order condition for the optimal production, gt(h+l) C'(y) -+-,X O, (7) where g'(h + 1) V'(h + 1)-H'(h + 1-[) is the mar- ginal future revenue subtracting the marginal Eckalbar makes the fixed ratio assumption with three reasons: (1) it is easy to work with; (2) it captures the spirit of the micro-level stocks literature; (3) it is in line with the fact.As already has been stated, Zhang (1989) replaces the fixed-ratio ad- justment with the non-linear adjustment function and obtains persistent cyclical behavior.6We can take a more rigorous approach to the multiperiod optimizing problem of the firm.We, however, do not do so for two reasons.First, we avoid to solve the complicated mathematical problem.Second, the gist of this paper is not to rigorously study the details of an inventory-holding firm but rather to reveal the role of a profit maximizing firm and the dynamics of the model.We use the simpler approach to inventory-holding behavior in order to highlight the purpose of the paper.Our ap- proach approximates it and captures its essential features.
There are some studies in a literature of optimal inventory theory that can be used to entail Assumption (3).cost for carrying inventories to the future period, and where A is a Lagrange multiplier associated with the non-negative constraint.An optimal condition for maximizing profit, (7), indicates that the cost of producing one additional unit today and storing it until tomorrow is not less than the revenue gained by selling one unit out of the inventory stock tomorrow.The optimal production depends on the relative magnitude among the initial level of inventory, h, the expec- tation on sales, s, and a level of inventory, de- noted by ho(s), that equates the marginal revenue to the marginal cost of holding inventory. 8In par- ticular, if h s > ho(s) holds, we have g'(h + 1) < 0 and C'(h + (h s)) > 0 for any h+ > h s.
In this case, the first-order condition, (7), leads to no production.That is, if the initial level of inventory is large enough, the cost of holding inventory is over the expected return so that the producer does not produce at all but liquidates stocks of inventory to meet demand for the con- sumption goods.On the other hand, if h-s < ho(s) holds, the optimal production, denoted by y* (s, h), satisfies the following condition: -/4'(h +/(% h) ).
Changes in an initial level of inventory and of expectation on sales alter the optimal level of production.A standard comparative statics exer- cise for the optimal production yields the follow- ing effects on the equilibrium production: Oy* H"-V" -1 < O---=-C,,+H,,_V,,<O, (9)   Oy* fill" + H"-V" C" >-1 as/3 > O< Os C" + H"-V" < < H" (10) Inequality conditions on Eq. ( 9) indicate that an increase in initial level of inventory reduces production but not the entire amount of the increase.The remaining amount is met with decreases in the optimal inventory carried-over. 9 The change in the expected sales shifts the gt(h+ curve and the C(y) curve, both of which affect the optimal production.If the shift of the curve dominates the shift of the C(y) curve (i.e., fill"> C"), changes in the optimal production is greater than the change in the expected sales, and vice versa.This is what the second inequalities in Eq. ( 10) indicates.
These considerations imply that the demand for labor has two phases: Ld(s,h) max{O,F-l(y*(s,h))}.We consider determination of actual transactions in the labor market and the goods market.We restrict our analysis to a "Keynesian" state.That is to say, the consumer achieves his desired trans- action in the goods market and cannot in the labor market while the producer can achieve his transaction in the labor market and cannot in the goods market.In order to highlight the endo- genous non-linearity of the model, we assume that the exogenous amount of labor supply, N, is not a binding constraint in the labor market.
The model functions as follows.At the start of a period, the producer holds an initial stock of inventory, h, and forms a subjective expectation Assumption 1(2) and (3) imply that g'(h+l)>0 for a small enough level of h+ 1, and g'(h+ ) < 0 for a large enough level of h+, and that g"(h+_l)=V"(h+)-H"(h+-h)< 0. Thus there is a level of inventory, ho(s), such that g'(ho(s))=O or v'(h0(s)) H'(ho(S)-).9We can see this by differentiating the intended inventory accumulation equation in (5) where y is replaced with y*(s,h) and h with the optimal inventory carried-over, h*+ (i.e., Oh*+/Oh + (-Oy*/Oh) 1). on current sales, s.Following the analysis above, the producer determines his desired demand for labor Ld(s,h) while the consumers offers a fixed quantity of labor supply, N. The consumer and the producer meet first in the labor market in which there exists excess supply.According to the min-rule, the demand side of the labor mar- ket determines the actual quantity of labor employed, L: L-min{N, Ld(s,h)} Ld(s,h).( 13) the following section, we assume a formation of adaptive expectations.That is, the producer adaptively adjusts his expectation according to a difference between demand for the consumption goods and current level of expectation, + . (s-  5)   where c is the adjustment coefficient and s+l denotes the expectation one period ahead.
After the labor market closes, the consumer chooses his demands for the consumption goods, S(L).Actual employment also determines the current production for output, F(L).The starting stock of inventory is a sum of the current pro- duction and initial inventory, h+F(L).This is the supply of the goods that we denote by yS(L).
The producer is assumed to hold enough amount of inventory so that the consumer always realize his desired demand for the consumption goods.Thus the actual sales, Y, is the demand for the consumption goods:

INVENTORY DYNAMICS WITH ADAPTIVE EXPECTATION
In this section we demonstrate that endogenous inventory cycles appear when the speed of expec- tation adjustment is varied. 1 The dynamic sys- tem that governs the expectation on sales, st, and the level of inventory, ht, is st+ st + (y*(st.ht))) st).
(16) Y--min{ yS(L),S(L)} S(L). ( At the end of the period, transactions complete and the economy is in a temporary equilibrium state in which the sum of actual purchases equals the sum of actual sales.A difference between the demand for the goods and output produced de- termines an actual level of inventory carried over to the next period.Moreover, the producer re- cognizes a difference between the expectation on sales and the actual demand.Consequently, the producer adjusts his expectations on sales in the following period.Hence the inventory accumulation and the expectation revision can be sources of dynamics of the model. Equation (2) governs the inventory accumula- tion process.In order to describe dynamics of the model, we need to specify how the expecta- tion is revised from one period to the next.In where the first is the expectation revised equation and the second is the inventory accumulation equation.In the following, we illustrate the dy- namic behavior of the model in phase diagrams as a first-order approximation and then analyze it mathematically.Before proceeding, we define an equilibrium state of the model that is a fixed point of the dynamic system, (16).DEFINITION An equilibrium state of the macro- model with adaptive expectations is a pair of ex- pectation and inventory, (s*, h*), such that y*(s*, h*) S(F l(y,(s*, h*))) and S (F-l(y (s*, h*))) s*.

Graphical Analysis
To make a graphical analysis of the behavior of inventory, ht, we find the locus of (st, ht) points along which the level of inventory is constant.
A time subscript, t, is attached to time-dependent variables hereon.

(17)
Determining the slope of the locus by differen- tiating Eq. ( 17), we find from ( 9) and ( 10) that it has a positive slope in the (st, ht) plane: To put it another way, output produced equals the quantity demand for (st, ht) on the constant inventory locus.Such an equality holds at a point where the production curve, F(L), crosses the demand curve, S(L).Both curves are increasing at non-increasing rates with respect to L. There will be no intersection, one, or two depending on exogenously determined parameters like prices, wage, autonomous demand, consumer's charac- teristics, properties of the production function, etc.
We make the following assumption to ensure the intersections.
ASSUMPTION 2 The F(L) curve intersects the S (L) curve twice.
Since F(0)--0 < S(0), Assumption 2 implies that the production curve crosses the demand curve from below and then from left as L increases from zero to infinity.We denote the first intersection by (yl, L1) and the second intersec- tion by (y2, L2) where Yi is output produced with L; (i.e.Yi F(Li) for i-1,2).At these points, the following inequality conditions hold: F' (L) > S (L) forL=L1 and F'(L) < S'(L) for L L2.

Oh
Oy*/Oh (19) We differentiate the second equation in dynamic system (16) and then transform the resultant Oh, Since the second factor, Oy*/Oht, is negative by Eq. ( 9), the sign of Eq. ( 20) depends upon the relative magnitude of the marginal product, F', and the marginal propensity to demand with respect to employment, S'.By Assumption 2, we have F'(L) > S'(L) for L=L1 and F'(L) < S'(L) for L L2. Thus h, + < ht if and only if (s,, hi) lies above the locus producing yl, while ht + > ht if and only if (s,, h,) lies above the locus producing Y2.
We can also determine a locus of (st, ht) along which the expectation on sales is constant and will call it the constant expectation locus.Following the first difference equation of the dynamic system, (16), the locus satisfies S(F-(y*(st, h))) st, As a result of Assumption 2, there are two con- stant inventory loci: one corresponds to the lower production, y*(s,h)=yl, and the other to the higher production, y*(s, ht) Y2.The constant in- ventory locus crosses the ht axis for h that satis- fies y*(0, h)=y (i=1,2).This intercept is positive or negative according to whether Yi is less or greater than the optimal level of production for s h 0. l which means that the expectation on sales is rea- lized.It is verified that this locus intersects the y*(st, ht)-O locus at a point (s,h) where s= S(0) and h satisfies y*(s,h)=O. 12y totally differentiating Eq. ( 21), we obtain the slope of the constant expectation locus: Os s,+,=s, S'Oy*/Oh O-]Os- S'   (22)   11Suppose that Y0 satisfies the optimal condition, C'(yo)= V'(yo)-S'(yo).y* (s,, h,) yo is a locus of (st, ht) starting at the origin, (0, 0) in the (s,,h,) plane.In Fig. 1, we assume Yl < Y0 < Y2.
Y*(St.ht )=0 Y*(St, ht )=Yl ' y*(st, ht )=Y2 where the first factor is negative but the sign of the second factor is ambiguous.F'/(Oy*/Os) is a reciprocal of the second equation in Eq. ( 12).It is a slope of the L Ld(st, ht) curve that, for each level of employment, L, measures what the ex- pectation on sales would have to be for the pro- ducer in order to choose that level of employment.
The constant expectation locus has a negative or positive slope according to that the L--Ld(s, ht) curve intersects the s=S(L) curve from below (i.e.F'/(Oy*/Os)>S') or from left (i.e.F'/(Oy*/Os) < S' ).Moreover, even if it is positive- ly sloped, the constant expectation locus is flatter than the constant inventory locus. 13To examine the dynamic behavior of st, we differentiate the expectation adjustment equation to obtain o(,+, ,)
We plot possible shapes of the constant inven- tory locus and the constant expectation locus in Fig. in which F'/(Oy*/Os) > S' is assumed.An intersection of two loci is an equilibrium state.There are two equilibrium states that we label el, and e2, respectively.At el equilibrium state the lower production, y, takes place (i.e., y*(s*l,h*)= y) while at e 2 equilibrium state the equilibrium production, Y2, takes place (y*(s],h])=yl).

Arrows in Fig. indicate possible movements of
trajectories generated by the dynamic system, ( 16).We can see that e 2 equilibrium is a saddle point and hence unstable except one stable path.As is seen below, the stability of el equilibrium depends on particular values of the adjustment coefficient, c, and of the inventory-expectation ratio,/3.

Stability Analysis
We analyze the local stability at each equilibrium mathematically.The Jacobian matrix, which is obtained by a linear Taylor expansion of the sys- tem (16) evaluated at the equilibrium point, is (24) The determinant, the trace, and the characteristic equation of ( 24) are, respectively, as follows: (25) It follows that the characteristic roots are A,,2 1 / 2 tr J(c) + x/-D(c) where D(c0 det J(c0- 1 (tr J(c0) 2 is the discriminant of the characteris- 4 tic equation.Depending on the sign of the discri- minant and on whether the modulus of the characteristic root is greater or less than unity, the trajectories diverge or converge.Next theo- rem confirms the graphical intuition that e2 equi- librium is a saddle point.

THEOREM
e2 equilibrium is a saddle point.
As can be seen in Eq. ( 22), the slope of the constant expectation locus is either negative or positive according to whether F'/(Oy*/Os) is greater or less than S. By Eq. ( 1), S=c, the marginal propensity to consume that is assumed to be constant.Both of F and Oy*/Os depend on the inventory-expectation ratio, /3.To emphasize the dependency of Oy*/Os on the value of/3, we denote Oy*/Os by f(fl).Returning to Eq. ( 10), we define two functions of /3: ()=H"(h+y*- (1 +fl)s) and rl(/3)=C"(y*).Since positive pro- duction takes place for /3=0, (0)=0 < r/(0).If we assume that these functions intersect only once, say, for = ill, we then have f(flt)= and '(flt (26) which leads to f'(fl)>0 for fl--fll" We assume that this positive relation in the vicinity of fll holds globally: ASSUMPTION 3 f'(fl) > 0 for all/3>_ 0, and there is a value of the inventory-expectation ratio, /3*, that satisfies f(fl*)= F'/S', where F' and S' are evaluated at e equilibrium.
Theorem 2 below states that e equilibrium is locally stable if a value of the adaptive coeffi- cient, c, is confined to an interval, (0, 1), and the inventory-expectation ratio,/3, is not so large.THEOREM 2 For fl < fl*, equilibrium is stable.
These lemmas imply the following theorem.
THEOREM 3 Given >_/3", el equilibrium is lo- cally stable for 0 < c < Cl(/3) and unstable for > Proof For c>Cl(/3), Lemmas and 2 imply D(c0 > 0 and det J(c0 > 1.The characteristic roots are complex conjugate and their moduli are greater than unity.Thus the trajectories are oscil- latory divergence.By the same token, Lemmas land 2 imply that D(c0 < 0 and detJ(c0 < for 0 < c < c0(/3) and that D(c0 > 0 and det J(c) < for c0(/3) < c < c1(/3).Thus for 0 < c < c(/3), the real roots are positive and less than unity, and the modulus of the complex root is less than unity.Hence the trajectories monotonically or oscillatory converge to e equilibrium.7q We turn to the issue of cyclicity in a case where e equilibrium is unstable.The following lemma is a truncated version of the Hopf bifur- cation theorem.s LEMMA 3 Let the mapping x + G(xt, oz), xt e R2, oe e R, have a smooth family of fixed points x*(c) at which the eigenvalues are complex 14As can be seen in Eq. ( 9), a sign of '(/3) depends on the signs of third-order derivatives of the functions involved.If (0@/3) (Oy*/Oh) is negative but its absolute value is small, or positive, we can have '(/3)> 0.
where det J(0) < and det J(1) > are shown in Lemma 1.The module crosses the unit circle with non-zero speed.This covers the assumptions of the Hopf Theorem.Hence given /3>/3", a Hopf bifurcation occurs at c c(/3).

F1
where y. is a first-order partial derivative of y*(s,h) with respect to the ith argument, and y. is a second-order partial derivative.Taking the Taylor expansion of the dynamic system (16) under Assumption 4, we have By Eq. ( 27) and Lemma 2, we can depict a parameter space of the inventory-expectation ra- tio,/3, and the adaptive coefficient, c, as Fig. 2.17 c0(/3) is a boundary between a region for real roots and one for complex roots, and c1() is a boundary between a stable region and an unstable one.

Stability Index
Taking account of the higher-order terms in the Taylor expansion of the dynamic system, we can compute the stability index of the limit cycle ob- tained in Theorem 4. To this end, we simplify the model to avoid lengthy calculations and then to make a change of coordinates so that the dynamic system is of the form provided by Wan (1978, see the formulation on p. 168).

CONCLUDING REMARKS
This paper extends the Eckalbar's disequilibrium macro-model.Linear behavioral functions are re- placed with non-linear conventional functions that are based on individual's intertemporal opti- mizing behavior at microeconomic level.The dynamic system consists of the inventory accu- mulation equation and the adaptively revised expectation adjustment equation.It is demon- strated that the model endowed with a large stock-expectation ratio generates endogenous in- ventory oscillations.These results suggests that the disequilibrium non-linear dynamics may provide useful explanations for irregularities that are observed in a macro-time series such as those of the real GNP, the unemployment rate, and the inventory investment.
to participants in seminars at University of Southern California and the Savings Economy Research Institute.All remaining errors are my responsibility.
Taking the Taylor expansion of the dynamic system, (16), yields where J is the Jacobian matrix and 0 2 is the higher-order terms.Elements of J are given in Eq. ( 29) in the text.The determinant and trace of the Jacobian matrix, J, are where A 1 / 2 tr J(al) and B v/det J(al) 1 / 4 (det J(al))2.