Simulating the inventory cycle

Journal of Economic Behavior and Organization 21 (1993) 147-179. North-Nolland

Simulating the inventory cycle Michael

C. Lovell*

Wesleyan University, Middletown, CT, USA Received August 1991, final version received April 1992

Simulations of the inventory cycle utilizing a model based on 84 interacting firms in 21 industries reveals that the ratio of the variance of sales to the variance of output is an unreliable indicator of how seriously firms attempt to smooth production or whether such efforts, in the aggregate, even out economic fluctuations. It is also shown that the ‘two slow’ estimates of the speed of adjustment reported in many econometric studies of United States data may result in part from aggregation bias rather than from limitations of the flexible accelerator model itself.

1. Introduction This paper analyzes inventory fluctuations utilizing a d&aggregated version of the macro models developed by Eric Lundberg (1937) and Lloyd Metzler (1941). Their studies neglected complications of aggregation by relying as a matter of analytic convenience on the assumption that the behavior of a single representative economic agent could be ascribed to the aggregate data for the entire economy. Also, the single firm constituting the economy did not hold stocks of purchased materials. In contrast, the research strategy employed in this paper allows us to consider quite complicated modes of behavior by a host of interacting economic agents, The simulations are based on a 21 industry multi-sector model; there are four firms in each industry, and each firm holds stocks of 21 different inputs as well as inventories of final product. Further, non-negativity constraints on output and stocks are explicitly taken into account. After explaining the details of the multi-sector model in the next section, this paper reports what happens when it is used to analyze the following two issues: (1) One class of questions concerns the types of dynamic behavior generated by this multi-sector model of the inventory cycle. For what values of the Cor~e~~o~de~ce to: Michael C. Love& Department of Economics, Wesleyan University, Middletown, CT 06457, USA. *I am indebted to Alan Blinder, Riccardo Fiorito and Louis Maccini for helpful comments on an earlier draft of this paper. George I. Treyz kindly provided in machine readable form the aggregated input-output matrix used in this study. 0167-268I/93/~6.~

0 1993-Elsevier

Science Publishers B.V. All rights reserved

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(2)

A4.C. Love& Si~ul~ti~~ rhe inventory cycle

parameters will the system be stable? Is it likely to generate cyclical movements or will adjustment to equilibrium be monotonic? These questions were considered by Lundberg and Metzler for their aggregated model, but the issues are more complicated in a multi-sector environment. For example, the multi-sector system may break down because of contagious stockouts: When one firm runs out of a critical raw material and has to halt production, it may for a time be able to satisfy its customer’s needs from finished goods inventory. Once its finished goods inventory is exhausted, its frustrated customers can attempt to source inputs from other vendors; and they can draw on their stocks of purchased materials; but eventually a shortage of critical inputs may force firms to shut down. Thus such questions as stability become much more complicated when they are studied in a multi-sector environment subject to inequality constraints. A second class of questions focuses on the implications for econometric research of aggregation and inequality constraints. Will distorted parameter estimates by obtained when aggregate data generated by this multi-sector economy are subjected to econometric investigation? For more than three decades economists have attempted with very limited success to estimate the effect of monetary policy on inventory investment, to determine whether inventory movements are influenced by price hedging and speculation activity, and to test the hypothesis that production smoothing considerations impact inventory behavior.’ It is an understatement to say that we are a long way from resolving these issues.* Many empirical studies of inventory data for the U.S. economy, based on one variant or another of the flexible accelerator model, yield parameter estimates implying unreasonably slow speeds of adjustment. It is conceivable that such anomalies arise from aggregation distortions and the neglect of inequality constraints? Section 4 reports on the degree of success achieved in using standard econometric techniques in attempting to uncover the underlying structure generating the data used in the simulation experiments.

These are interesting

questions, but could they be resolved analytically

‘Brennan (1958) addressed the question of price hedging; Mills (1962) considered the role of price changes and production smoothing; Love11 (1959) presented non-encouraging results for the production smoothing hypothesis but failed to report his unsuc~ssful attempt to link desired inventory holdings to interest rates and anticipated price movements. ‘The question of whether firms smooth production has recently become a matter of debate, engaging the best efforts of a number of economists, including Blinder (1981), Fair (1971, 1989) and Ghali (1987). Whether and how central banking policy influences inventory investment is an important issue that has not attracted much attention, possibly because of the difficuities encountered in publishing negative results running counter to conventional wisdom. Irvine (1981) is the only investigator who has consistently reported the expected effect.

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rather than by resort to simulation ex~~ments? Lovell’s (1959, 1963) study of a multi-sector input-output model was limited to finished goods inventory in order to make the model analytically tractable. Foster (1964) analyzed the stability of an alternative multi-sector model with purchased materials stocks but no finished goods inventory. While these analytic results helped guide this research project, the results reported in this paper are based on a quite different research strategy. Resort is made to computer s~ulation in order to look at the more interesting but analytically intractable case of a multi-sector model holding both purchased materials and finished goods inventory.3 While the simulation strategy utilized in this paper allows us to consider a much richer model than earlier theoretical studies of the inventory cycle, this investigations does share certain limitations with the inventory cycle literature in the ~udenberg-~etzler tradition and the vast majority of recent empirical studies of inventory behavior and production smoothing; in particular, speculation in stocks is not considered and it is assumed that firms adjust to surprises in demand by changing quantities rather than price. The first section of this paper explains the structure of the model used to generate the experimental data. Section 3 looks at the type of cyclical behavior generated by the multisector model. Section 4 examines how successfully simple econometric techniques work at uncovering the structure by which the data are generated. Conclusions are presented in section 5. 2. The structure An overview of the decision sequence of the purchased material plus finished goods inventory model is provided on table 1. After reviewing in detail the steps of this sequences, the discussion will focus on certain basic considerations underlying the structure of the model and the appropriateness of the assumptions. While it may be objected that many of the assumptions of the model are obviously ‘ad hoc’ certain key elements of the inventory model can be derived from the assumption of profit maximization - thus the model can be said to involve a degree of limited rationality. 2.1. The production-order sequence The orders received at each suppl~ng firm Row in part from the orders placed by other firms to replenish their stocks of purchased materials. In 3Bivin (1986) reports input-output simulations focused on the task of estimating the multiplier effects of shifts in final demand in a model in which both purchased material stocks and finished goods inventory were determined by a flexible accelerator. While Blinder (1981) used simulation on a much smaller scale in his study of the implications of (S,s) inventory policy, the strategy is much more commonly employed in studies of how firms should behave in the operations research and management science literature, such as the Marilyn McCIelland, Southard and Wagner (1988) analysis of ‘inventories and lot-size strategies in an MRP environment’.

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the inventorycycle

Table 1 Decision sequence. Each firm may recalculate S,s points for purchased materials stocks on the basis of forecast sales. P(Recalc), a parameter of the simulation, is the probability that a firm will reevaluate its stocking policy in the current period. Calculation of the (S,s) points may be based on the Economic Order Quantity lot-size formula, reflecting the appropriate tradeoff between ordering costs versus carrying costs. Recalculated (S,s) points are not implemented unless they change by more than 20%. Orders are sent to suppliers for each stock that is below the s trigger point; enough is ordered to rebuild stocks to the upper target level S. The supplier is chosen on the basis of reliability ratings - All potential suppliers are rated each period, if a supplier delivers on schedule, its reliability rating goes up; if it fails to deliver, its rating drops substantially. Production is scheduled for each period on the basis of customer orders, complicated by several considerations: (a) Desired finished goods inventories are a linear function of ‘normal sales’, which are calculated utilizing the exponential smoothing formula discussed in much of the management science literature. (b) Targeted output exceeds or falls short of orders on hand by a fraction of the gap between the desired stock of finished goods inventory and the stock left over from the preceding period. (c) Production smoothing considerations may cause actual output to change by only a fraction of the gap between the output level required to achieve targeted inventories and last period’s output level. (d) End of period inventory equals the beginning period level plus the excess of production over sales. Output is shipped to customers on the basis of orders received in step no. 2, if available; additional shipments meet consumption and final demand from the government. The interindustry customers receive first priority if the sum of current output plus finished goods inventory held over from the preceding period falls short of what is required to fully satisfy demand.

addition to the flow of inter-firm orders, firms receive orders reflecting the exogenously determined purchases of the government sector and the endogenous purchase decisions of consumers. Each sector’s share of total government purchases varies randomly from one period to the next. Total consumption equals the marginal propensity to consume times gross sales (including inter-industry flows rather than just final output) and is prorated among all the supplying firms producing the same commodity. The marginal propensity to consume is one of a number of parameters that can be varied from one simulation to the next. The timing of the flow of orders each firm receives is influenced by the stork-holding decisions of its customers. Each firm decides how much to order of each input from the appropriate supplier in accordance with the standard (S,s) Economic Order Quantity (EOQ) decision rule: when a stock is observed to have fallen below the critical lower value s, an order is placed to rebuild the stock to the upper trigger level S. The order for each input could be pfaced with any of the four potential supplying firms in the industry producing that input. In accordance with a modeling strategy developed by Barbara Bergmann (1989), the purchasing firm scores potential suppliers for delivery reliability, upping the rating for prompt delivery and subtracting

M.C. Lowell, Simulating the inventory cycle

1.51

penalty points when orders are not filled. If it finds an alternative producer that appears to be signi~cantly more reliable than its customary supplier, it will shift to the new source. And orders are not backlogged, if they cannot be filled from finished goods inventory in the current period, they are lost forever.4 Thus the penalty imposed on a firm that runs out of finished goods inventory may involve more than the loss of current sales because a customer firm, angered by the unreliable behavior of the supplier, may exercise its option of shifting to an alternative source, if one exists with a higher reliability score. Such ‘source shifting’ may become epidemic if shortages frustrate many customers. How frequently source shifts occur will be tabulated for each simulation. Each firm sets its output to meet projected sales plus any excess of targeted inventory over the finished goods inventory inherited from the preceding period, but tempered by production smoothing considerations5 This basic framework, but with innumerable variations, has been used for thirty years in many empirical studies of inventory behavior, including Love11 (1961) and Fair (1989). Each firm’s normal tinished goods inventory at the end of period t, H:, is assumed to be linearly related to normal sales, X:.

Normal sales in turn are obtained by smoothing incoming orders with the standard exponential smoothing formula.6 X:=80rders,+(l-B)X:_,,

05@el.

Several factors may prevent normal finished goods inventories from being realized. First, the firm may desire to make only a partial adjustment (6) of the inventory stock to the normal level during a single planning period.

Hf-IT_,

=6(H;-H,_,),

0<651.

However, this standard assumption will be qualified in recognition of the fact that output must be non-negative, which means that inventories cannot be 4The alternative strategy of allowing orders to be backlogged would open up a number of complexities, including the need for rules governing the circumstances under which customer firms may cancel orders that their vendors have not filled promptly and the possibility that customers experiencing order delays may double or tripie order in an attempt to receive prompter service. These complications were not implemented by Bergmann (1986) and have not been introduced in this study. 5A convenient summary of the firm’s decision making process is provided by the annotated Pascal code procedure (subroutine) for the production decision presented in Appendix A. 6The Holt-Winter expo~enti~ smoothing procedure discussed in many management science and production scheduling textbooks is designed to take into account seasonal fluctuations in sales, a complication that is neglected in the simulations reported here; cf., Holt, Modighani, Muth and Simon (1960, ch. 14).

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M.C. Love& Simulating the inventory

cycle

liquidated at a faster rate than shipments. And while the achievement of the desired level of finished goods inventories would require production of this change plus anticipated sales X:, dynamic costs incurred in changing the level of production may lead the firm to attempt to smooth production, adjusting output to a value intermediate between the above value and that of the preceding period. If so, we may have: Q,=(l-z)(H:‘-H,_,

+X;)+zQ,_,,

OS;z
(4)

Quite apart from production smoothing considerations, supply bottlenecks or .delivery delays may cause an output shortfall. The model incorporates the rigid Leontief input-output assumption that production is subject to constant returns to scale, with the demand for each input being a fixed proportion of the level of output; that is to say, the requirement for the ith input in the production of output Qj of product j is UijQj, where aij is the amount of input i required in the production of a unit of good j.? The simulations to be reported later in this paper will invoke the strong uniformity assumption that such basic parameters as fil, 0, 6 and z are the same for all firms in the economy; and the Uii will be the same for all firms in each industry, although there will be inter-industry variation. The strong unifo~ity assumption, which was also invoked in much of the theoretical work by Love11 (1963) and Foster (19641, is convenient but not essential for the analysis; but empirical studies of industry behavior, including the recent study by Fair (1989), have not as yet provided significant evidence about the direction of inter-industry variations in the parameters.8 2.2 Resetting the (S, s) trigger points As every business school student learns, the Economic S-s, is given by equation S-s=(2a,U”/a,)1’2,

Order Quantity, (5)

where a, is the fixed cost of placing an order, a, is the cost of carrying a unit of inventory for one period, and U” is the normal rate of utilization of the item. The specific values of the trigger points governing a firm’s purchased material stocks will be calculated separately for each input, depending on the importance of the input in the firm’s production process. The firm producing 7While the assumption of fixed technological coefficients helps keep the memory requirements of the program within bounds and is simple to program, the methodology employed in this paper could in principle be extended with a more sophisticated program and computer to incorporate substitution in the production process. *A major advantage of the strong uniformity assumption is that it makes the task of estimating the parameters of the model from aggregate data unambiguous when the time comes to test out how well econometric techniques recover the undertying structure of the model.

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Lovell, Simulating the inventory cycle

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good j has normal utilization rate for input i, Vi, that is related to normal orders XT [dete~ined of eq. (2)] by

where aij is the Leontief input-output coefficient reporting the quantity of input i typically used in the production of a unit of output j. Under these assumptions the Economic Order Quantity is (S-S),

=(2a,U~jXjn/Ul)i’2-

(7)

In the simulation experiments it will be assumed that the ratio of ordering to carrying costs are the same for all inputs; defining a simulation parameter Smax=(2%/%)“2 simplifies the expression for the optimal order quantity to (S--s),=

S aijXjnl’? inBX

(8)

The lower trigger point s, often called the ‘safety stock’, recognizes that orders may not be tilled instantaneously and provides insurance against stock-outs. Its optimal value should reflect an appropriate balance of the carrying costs incurred in holding the safety stock (analogous to the insurance premium) against the loss that would be incurred if a critical shortage disrupts production - this is likely to involve the possibility of occasional stockouts rather than 100% protection against the possibility of the exhaustion of purchased material holdings.’ The lower si trigger point, governing the size of the ‘safety stock’ that a firm in industry j will hold of input i, will be assumed to be linked by a proportionality factor Sminto the normal utilization rate: si= S UJ

(9)

min =

S min

~ijX~.

W)

Under the strong uniformity assumption, the simulation control parameter Sminwill be the same for all 84 firms. However, the average level of stocks of each input is likely to vary among firms in the same industry because they may have different levels of normal orders; and they will vary between inputs for the same firm because of differences in utilization rates as determined by the aij. A change in the general level of normal sales and/or interest rate and other ‘Obviously, if demand is potentially unbounded (e.g. log-normally distributed), no finite level of stocks would provide 100% protection against stockouts if suppliers cannot be counted upon to deliver an unbounded quantity of inputs instantaneously, which would obviate the need for any safety stock. The tradeoffs are discussed in some production scheduling textbooks; e.g. Turbin and Meredith (1988, pp. 621-622, 624-630).

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Love& Simulating the inventory cyde

carrying costs may imply that the ($8) points should be recalculated; i.e., the simulation parameter S,,, should be reset. While the simple mathematics involved in recalculating the (S,s) trigger points is a standard part of business school curriculum, the question of how frequently the trigger points should be reset appears to be neglected in representative business school texts on production scheduling and in the management science literature as well as in econometric studies of inventory behavior. Although the cost of recalculating may be trivial, substantial expense might be involved in adapting storage facilities to implement revised (S, s) trigger points.1° If the (S,s) points are rarely reset, the size of purchased material stocks will be largely insulated from changes in demand conditions and carrying costs. This has an important policy implication, for it would mean that stocks would be insensitive to changes in monetary policy, implying that the inventory cycle is less likely to be amenable to stabilization by central bank intervention. And it would also mean that the level of stocks would be largely unresponsive to output changes over the course of the business cycle, except in so far as cyclical forces influenced the rate of birth and death of firms and/or lead to a clustering of delivery delays and stockouts at certain stages of the business cycle. As part of the simulation exercise firms are allowed to review the appropriateness of their current (S,s) trigger points in the light of changing market conditions. The question of how the inventory cycle is influenced by the frequency of recalculating the Economic Order Quantity (EOQ) is investigated experimentally by changing the simulation parameter P(Recalc) specifying the probability that a firm will recalculate its (S,s) points for the coming period; however, the recalculated trigger points were implemented only if they deviated by at least 20% from the old values. When the probability of recalculation is set close to unity, a sizable number of firms may recalculate the (S,s) trigger points over the course of the business cycle in an effort to fine-tune their inventory policy to changing economic conditions. With P(ReCalc) = 0, the (S, s) points are never recalculated. iI Under the assumption of fixed proportions, the input-output coefficients used in calculating the normal requirements for each type of input when setting, (S,s) trigger points will also link the firm’s rate of utilization of each ‘OFor example the flexibility of a neighborhood gas station in adjusting the reorder quantities for gasoline is Ii&ted by the size of its storage tanks. The possibiI~ty of recalculating the (S,s) trigger points is considered by Biinder (1981); in contrast, both Caplin (1985) and Mosser (1988) assume the (S,s) points remain fixed throughout the period of analysis. “*Obviously, tliis is only one of a variety of strategies that might have been used to capture the fact that firms do not continuously implement revisions of the (S,s) trigger points. For example, it could have been assumed that firms recalibrate the points quarterly or annually rather than on a random basis. I have not been able to discover evidence in the literature about what firms actually do. That high priority should be assigned to closing this gap in our knowledge is established by the simulations reported later in this paper, for it turns out that the frequency of revision has a critical influence on the stability of the economy.

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input to the level of output, where output in turn is determined by eq. (4), unless disrupted by critical shortages of an input. The total orders received by each firm is the sum of the orders of all firms utilizing its product plus ‘final demand’, where final demand consists of government purchases, which are exogenous, and consumption, which is linked to total output by the marginal propensity to consume. Each firm fills orders to the extent permitted by its stock of finished goods, but must use a priority allocation system if there is a shortfal1.l’ 2.3. Recapitulation In reviewing the structure of the multi-sector model it is useful to note that the behavior o the system is subject to a number of inequality constraints. The adjustment process may be slowed in recession because output cannot be negative. I3 And as Hicks (1949) and Leontief (1953) recognized, stocks cannot be liquidated faster than sales, which may have a profound effect on the way the economy recovers from cyclical reversals in the pace of economic activity. And the boom may be inhibited by the fact that stocks of purchased materials and finished goods can not be negative, which means that production goals may not be achieved when delivery lags lengthen and stocks drop further than anticipated below the lower s trigger point. If stockouts do occur, production may have to be cut back below target, causing a ~run-down of finished goods inventory below the planned level. If finished goods inventory are completely liquidated, sales will inevitably fall short of orders, causing problems for the firm’s customers. These complications were left out of the Lundberg-Met~ler analysis of the inventory cycle and could not be dealt with by the linear models used by Love11 (1962) and by Foster (1963) in their derivation of stability conditions for their disaggrei2Priority was assigned to the ith firm in the jth industry by a scheme that minimized the occurrence of economic gridlock: the first firm in industry one received highest priority, the first firm in industry two second priority, and so on; after one firm has been served in each industry, firm two in the first industry was served, etc. Final demand of consumers and the government is met only after all inter-industry demand had been satisfied. In a preliminary stimulation study presented at the 5th International Symposium on Inventories, Love11 (1988), I found that economic gridlocks occurred with great frequency - the system locked up because no firm could obtain all the inputs required for production. In that model each firm held the same quantity of each input without regard to the rate of utilization and the priority scheme supplied all firms in the first industry before serving any in the second, etc. In subsequent research it will be interesting to allow substitution among inputs and competitive bidding in the event of shortage. 13While Hawkins and Simon (1949) demonstrated that the static input~utput model would not generate negative outputs for any non-negative vector of final demand with reasonable values of the aij, this obviously does not imply that a dynamic input-output model should be constructed without regard to the restriction that outputs must be non-negative in accordance with the fact that production processes cannot be operated in reverse, automobile factories using cars to produce steel and steel mills inputting steel to produce coal and iron ore. Some solace may be derived from the thought that if a linear model is shown to be stable, then the inequality constraints may not be encountered for sufftciently small disturbances.

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gated inventory cycle models. And the implication of delivery delays were excluded by Chaplin (1985) in his analysis of (S,s) inventory policy. Before proceeding it is necessary to reflect on the obvious fact that the model to be used in the simulations is subject to a number of limitations, many of which are shared with earlier studies of inventory fluctuations. In particular, many of the assumptions are open to the charge that they are ‘ad hoc’, although some may be defended on the grounds that they are consistent with current practice, that they are covered in the standard business school curriculum, and/or that they are derivable from the assumption of optimizing behavior, at least under rarefied conditions. To illustrate, stocks of purchased materials ~~0~~~be determined by an (S,s) decision rule, at least if demand is identically and inde~ndently distributed. An observed order quantity S -s, net of the complications of delivery lags, implies, via eq. (5) that the ratio between setup costs (a*) and inventory carrying costs (aI) satisfies the condition Q/al = (S - s)2/2U”,

(11)

at least if the points have been appropriately updated to reflect current cost conditions and sales volume. Thus a specified value of the simulation parameter S,, of eq. (8) can be rationalized as consistent with the assumption of cost minimization. However, optimality condition (11) is derived under the assumption that the orders a firm receives are independently and identically distributed, which would be violated if sales are subject to systematic seasonal or cyclical forces or if contagious stockouts are generated by the simulation. And while stock holdings are initialized at the start of the simulations by assuming they are randomly distributed over the interval (L&s), that assumption may be reasonable in terms of the argument of Caplin (1985) but can break down if widespread delivery lags develop. Production decision rule eq. (4) may be harder to defend, although it is in the spirit of the Halt, ~odigliani, Muth and Simon (1960) derivation of linear decision rules from quadratic cost functions. As Holt and Mo~glian~ (1961) demonstrated, a rich family of such rules can be derived from the assumption of optimizing behavior - that is to say, the assumption of optimizing behavior does not impose substantial restrictions on the form of the linear decision rule. And the Holt-Modigliani assumption that there are costs in changing the level of output reconciles the production smoothing complication with constant returns to scale.14 However, the optimaI linear 14Production smoothing behavior may be the consequence of a convex cost function, as assumed in the classic study of Edwin Mills (1963). The analysis of Holt, Modigliani, Muth and Simon (1960) and by Holt and Modigliani (1961) explained how production smoothing could arise with a cost function that exhibited constant returns to scale in Q, augmented by adjustment costs which were assumed to be proportional to the square of the change in the level of output, (Q, -Q+ #.

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decision rule will generally involve a weighted average of sales anticipated for a number of future periods - the time profile of future demand may be important. This complication is suppressed in many empirical studies of inventory behavior by the assumption that expected sales can be summarized in a single number, either because of a short horizon or because of a fixed time profile of anticipated future sales. The model of firm behavior used in this paper, like many other studies of inventory behavior, suppresses the role of price adjustment. It ignores the possibility that a firm may respond to inventory imbalances by adjusting price. The basic (S,s) rule governing stocks assumes that prices are not subject to fluctuations; that model neglects the possibility that a firm having difficulty obtaining delivery of desperately needed materials might offer to pay a premium for expedited delivery. Also, the linear production decision rule neglects the possibility that if finished goods inventory proves excessive, the firm might cut price. I5 More than this, the multi-sector models incorporate the assumption of fixed proportions rather than substitution in the production process. While the simulation model is open to the objection that the assumptions are ad hoc and defensible as representing ‘limited rationality’, it should also be noted that the modular nature of modern top-down structured programming computer languages means that in future research the simpli~ed decision rules invoked in this study could be replaced by more complex behavioral assumptions, subject to the constraint of computer capacity. While the present model is suf~ciently complex for exploratory analysis, at a later stage the structure can be modified to allow for price adjustments, for substitution in the production process, and other refinements and complexities. 3. Dynamics What types of dynamic behavior will be generated by the multi-sector input-output model? In their classic aggregative studies of the inventory cycle, Lundberg (1937) and Metzler (1941) considered two fundamental types of dynamic developments, focusing attention on the following questions: - Under what circumstances will cyclical rather than monotonic movements be generated? - For what values of the parameters will the system explode rather than converge to equilibrium? ISIt is not a difficult exercise to make the price of the firm’s product endogenous by introducing a suitably specified demand equation into the conventional linear decision rule production smoo~in~ model developed by Holt et a1. (1960); they did not choose to introduce this complication, perhaps because at the time they did not regard price adjustments as a realistic assumption about how manufacturing firms respond to demand shorfalls.

M.C. Love& Si~u~u~~nghe inventory cycle

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00

IO

20

30

40

50

60

70

80

00

70

80

90

rem Fig. 1. Simulation

no. 2.

25055

20050

8 0

15505

10000

5 00

10

20

30

40

55

60

Pm Fig. 2. Simutation

no. 28.

As fig. 1 ilfustrates, the multi-sector model is capable of generating cyclical behavior, although the fluctuations are not of the smooth sinusoidal form produced by the Lundberg-Metzler 2nd order non-stochastic linear difference equation model. Another experiment generated the pathological mode of behavior reported on frg. 2, In this ~rnu~at~o~ experiment the economy went on an inventory ~accumulation binge’ during the early period of the simulation; but after stocks had climbed to phenomenal levels output

M.C.

Lovell,

Simulating

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cycle

0 00

02

04

06

08

10

12

14

16

18

20

Fig. 3. Simulation no. 18.

suddenly collapsed into a prolonged crisis negligible output during which final demand was largely satisfied by running down excessive stocks of finished goods. When production finally started up again, the revival spread slowly because many firms produced by running down stocks of purchased materials rather than by acquiring inputs from other firms. These accumulation binges differed significantly from the symmetric sin-wave cycles of fig. 1. The case of ‘Economic Grid-Lock’ is displayed on fig. 3. Economic GridLock arises when spot-shortages become contagious. A firm failing to obtain inputs from its customary supplier may be able to maintain production temporarily by drawing on its safety stocks. It can try to shift to a new supplier, but this may be a vain effort if shortages are spreading throughout the supplying industry. If the shortage persists, the firm will eventually be forced to shut down production. For a time it may still be able to supply its customers from its inventory of finished goods, but eventually the firm that is frustrated by a shortage of essential inputs will be compelled to turn off the spigot to its own customers. When shortages spread in this way from firm to firm and industry to industry, the chain reaction can result in economic ‘Grid-Lock’, every firm being shut down because of a lack of at least one indispensable input.16 r6The case of economic gridlock is an obviously unr~listic outcome that the model is capable of generating It can arise because price adjustments are neglected, as in the vast majority of empirical studies of inventory investment and aggregate models of the inventory cycle. If gridlock were encountered in most or all of the simulations, it might be fair to conclude that a multisector model of the inventory cycle in the Lund~r~-Met~ler tradition is not viable. In

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M.C. Lmell, ~i~~lut~~g the ~n~e~~ry cycle

The type of dynamic behavior the model generates depends primarily on the settings of the various parameters of the model. In the next section we survey the limited extent of analytical derived knowledge about the behavior of multi-sector dynamic systems. Then we will review the results of 32 different experiments based on a wide range of values for the system’s parameters. Later we will report on how closely the results of the disaggregated system can be predicted by macro modeling. 3.1. Analytical results While Lundberg relied on simulations in investigating the inventory cycle, Lloyd Metzler derived analytically the properties of the second-order linear difference equation generating the inventory cycle. Metzler determined the range of values for the marginal propensity to consume and desired inventory coefficient for which the second order difference equation’s roots would be less than unity in absolute value, as required for stability, and when they would be complex numbers rather than teal, as required for cyclical movements. These results do not in general carry over to the disaggregated multi-sector model. Only limited analytical results are available for multi-sector models of the inventory cycle. Love11 (1963) reported a number of theorems for the special case of no purchased materials inventory (100% Just in Time) simplified by the exclusion of production smoothing and the neglect of inequality constraints; it turns out that with static expectations the model would be unstable for reasonable values of the Leontief matrix A=[uij] of inputoutput coefficients if firms attempt an immediate adjustment of inventories to their equilibrium level (a== 1 in eq. 3). Also, an increase in 6, in the marginal propensity to consume, or in any of the aij necessarily makes the system less stable.17 Furthermore, the model is necessarily unstable for any admissible magnitude of its parameters for expectations that are strongly rational in the sense that all firms correctly solve the model in order to determine precisely subsequent research I hope to construct a more elaborate model incorporating substitution in the production process, speculative inventory holdings, and a balance between price and quantity response to shifting demand conditions. 17The possibility that production smoothing would provide an alternate source of inertia was not explored, i.e., it was impli~~y assumed that r=O in eq. (4). I showed (1962) that the multisector linear inventory cycle model neglecting the complications of purchased materials stocks and non-negativity constraints would be unstable, outputs diverging further and further from long-run equilib~um, unless c/‘, the largest characteristic root of the input-output matrix satisfied the condition: c?‘<(2--6)/(2+6+2~,), where fi, is the marginal desired inventory coefficient and 6 the inventory adjustment coefficient. The 1986 21 x21 input-output matrix used for the simulations reported in this paper has a dominant characterstic root c&)=0.3878. This condition is clearly violated for Experiment no. 5, which nevertheless generates stable movements, thanks perhaps to the non-negativity conditions and production smoothing complications neglected in the derivation of the theorem.

M.C. Love@, Simulating the inventory cycle

161

their sales volume in the next period. ‘* Foster (1964) obtained rather different stability conditions for a model involving purchased materials stocks but no finished goods inventories; Foster invoked the ingenious assumption that expectations were only bounded-rational in the sense that firms were assumed to solve Leontiefs static input-output model to predict demand without attempting to take into account the dynamic fluctuations that would arise because their customers would be adjusting their stocks of purchased materials. While the analytical results derived by Love11 and by Foster for linear models are general in that they hold for any finite number of sectors, the stability conditions are not directly applicable to the computer model developed in this paper, in part because of the non-linear barriers imposed by the rest~ction that outputs and stocks can not be negative, in part because the simulation model incorporates both purchased materials and finished goods inventories, and in part because the simulation involved more elaborate forms of inertia and a more involved lag structure. Nevertheless, the analytical results do suggest two conjectures that may hold for the more complicated models used in the simulations: - First, the stability of the model will be reduced if either the marginal desired inventory coefficient or the marginal propensity to consume increase. - Second, stability is more likely if there is a increase in the system’s inertia resulting from a decrease in 6 [the inventory adjustment parameter of eq. (3)], an increase in r [the production smoothing coefficient of eq. (4)] or a decrease in 0 [the exponential smoothing coefficient of eq. (2)]. 3.2. Simulation results The essential features of 32 computer generated simulations are reported in successive rows of table 2.19 The first 10 columns of the table specify the parameter settings of the simulation, as defined by the glossary at the end of the table. And the experimental outcomes are summarized in the remaining entries in the same row. All of these simulations involved 400 periods - 100 years of quarterly data - except when cut short by Economic Gridlock.

I81 called these ‘perfect myopic’ expectations because the firms did not worry about the longer run implications that would follow if every firm held such expectations. ‘91t might be simpler to focus attention on a single simulation capturing certain ‘stylized facts’ about the inventory cycle without providing the details of the experimentation undertaken in order to &calibrate’ the model in order to generate the desired time-profiles. The calibration strategy would not reveal the sensitivity of the outcome to variations in the parameters of the model, which is a matter of primary interest. And it might not reveal whether there exist multiple sets of parameter values that would yield simulations consistent with the stylized facts.

j$, (3)

0.5 0.5 0.5 0.5

:: 0.5 0.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

:I:

0.5

0.5

MPC. Finished goods, purchased material interactions: 20 0.00 200.0 0.1 0.3 0.5 3.0 1.0 21 0.00 200.0 0.1 0.3 0.5 6.0 1.0 22 0.00 800.0 0.1 0.3 0.5 10.0 1.0 23 0.00 200.0 0.1 0.3 0.5 10.0 1.0

1.0 1.0 10 l:o 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

1.0

0.5 0.5 0.5

2.0 10.0 3.0

il:: 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

3.0

3.0

1.0 1.0 3.0

0.5 0.5 0.5

0.5

Purchased materials simulations: 17 0.30 200.0 0.1 0.3 18 0.30 200.0 0.1 0.3 19 0.30 200.0 0.1 0.3

0.3

(65)

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.0 0.8 0.5 0.5 0.5 0.5

simulations: 200.0 0.1 100.0 0.1 800.0 0.1 50.0 0.1 200.0 0.3 200.0 0.1 200.0 0.1 200.0 0.1 200.0 0.1 200.0 0.1 200.0 0.1 200.0 0.1 200.0 0.1 200.0 0.1 200.0 0.1

0.1

(Bd,

P(Ss R&al) (9)

0.5 0.5 0.5 0.5

0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.2 0.8 0.1 0.9

0.5

QExvon Smdoth (10)

726 9,869

0

11 11,114 0

:: 0 0 0 0 0 0 2

36 0 0 0 0 0

0

Source Shifts (11)

72 0 0 3 0 0 506 0 0 0 0 0 0 0 0

0

FinGood Exhausted (13)

0 1,617 46 0 22 84 807 12,423

141 13,841 864 13,679 0 208

559 187 187 193 208 461 452 653 85 718 9 38 575 7 173

187

Inout B&e (12)

Experimental outcomes

PM plus FG simulations (4 firms in 21 industries).

InputStocks t Prod ~Smooth Smin Smax (6) (7) (8)

.-____-.-...__I

0.3 0.3 0.3 0.3 0.3 0.7 0.0 1.0 0.1 0.3 0.3 0.3 0.3 0.3 0.3

Finished good 2 0.50 3 0.30 4 0.30 5 0.30 6 0.30 7 0.30 8 0.30 9 0.30 10 0.30 11 0.30 12 0.30 13 0.30 14 0.30 15 0.30 16 0.30

Benchmark run: 1 0.30 200.0

(11 G?

No. MPC

Finished -___

Experimental conditions

297

169 263 890

368

5.36% 100.00% 100.00% 100.00% 99.80% 9.85%

158 99.90%

100.5% 106.1% 104.1%

108.2%

92.3%

110.1% 95.3% 109.6%

99.2%

105.5% 107.0% 107.0% 107.3% 110.9% 108.6% 80.2% 107.5% 98.0X 102.8% 112.5X

107.0%

D dhltDUt/ Output &ales (15) (16)

99.96% 2,606 100.00% 297 100.00~ 297 100.00% 299 100.00% 346 loO.OOo/, 374 99.82% 125 100.00% 389 lOO.OO% 212 lOO.OQ”/, 326 100.00~ 263 100.00% 200 lOQ.00~ 348 112 100.00% 100.00~ 349

100.00%

Shiv/ Order (14)

_

5 12 B G-52

0 G-32 6

1 5

3 7

4 3 4 4

5c 3 3 3 3 6 Oe

3

Cvcle Char (17) --

P w.. a 8 b+ 4 il” %

3: h E: L “T= % 3 ” B B

0.00

conditions:

200.0 2,00u.o 200.0 200.0 200.0 200.0 200.0 200.0 200.0

0.1 0.1 0.3 0.1 0.1 0.1 0.1 0.1 0.1

0.5

0.3

10.0

20.0 20.0 10.0 10.0 IO.0 3.0 10.5 3.0 1.0

1.0 1.0 I.0 1.0 1.0 1.0 1.0 x.5

0.0 1.0

0.5 0.5

0.5

0.5

Expriment’s serial number Marginal Propensity to Consumea Intercept of desired inventory function [eq. (l)] Slope of desired inventory function [eq. (l)] Inventory Smoothing Coefficient [eq. (3)] Productibn Smoothing Coefficient [es. (4)] Parameter determining the Safety Stock feq. (9)l Parameter influencing-reorder quantity [eq.’ (d)jProbability that 3,s points recakndated Exponential Smoothing Coefhcient [eq. (2)]

0.5 0.5 0.5 0.5 0.5 0.5 0.5 0’5

0.3 0.3 0.3 0.3 0.3 0.3 Il:i: 7,623

5,547 3,900 886 129 561 0 0 583

444 455 122 7 36 0 501 0 9,973

7,665 5,669 1,419 147 643 0 0

13.76% 14.22% 99.12% 3,587 99.85% 1,008 99.62% 3,454 129 100.00% 1OO.W~ 100.00% 229 129 6.86%

11 ‘Source Shifts 12 Input Bottle 13 FinGood~xhausted 14 Ship/Order 15 a-output 16 o-output/a-sales 17 Cycle char

Number of times firms shifted to alternative supplier Number of times iaput bottlenecks forced production to fall below the planned level. Number of times finished goods inventory exhausted Average shipment to sales ratio Standard deviation of outputb Ratio of output to sales standard deviation Characteristics of the business cycle: integer- number of severe cyclical downturns U- unstable B- Accumulation binge and crash G-t-Gridlock terminates all output by period t r- Erratic behavior “This is the margina propensity to consume ant of total output, including interma~a~ product. bNot calculated for simulations terminating prematurely because of grid-lock.

Exuer~mental outcomes:

9 P(SsRecal) 10 0

62 7 Smin 8 Smax

Experimental 1 No. 2 MPC 3 PO

25 0.00 26 0.10 27 0.10 28 0.20 29 0.10 30 0.10 31 O.&I 32 0.10 -I._____ Glossary

24

97.1% 108.7% 108.2% 87.2% 108.6% 87.2%

G-61 G-62 8+B 6 4+B e e e G-34

As the table indicates, the deferent experiments resulted in quite different outcomes in terms of how often tirms decided to shift the source of purchased material inputs because of the unreliability of supplyj~~ firms (column ll), the frequency with which input bottlenecks hampered production (column 12), the extent to which producers were unable to fill the orders of their customers because of finished goods inventory stockouts (colunms 13 and X4), and the standard deviation of output (column 15). ~orn~ar~son of experiments no. 1 and no. 2 suggest that an increase in the marginaf propensity to consume has a decidedly adverse effect on the stability of the economy - gOoutputshoots up and a number of supply and bottleneck problems are encountered. Experiments no. 3, no. 4 and no, 5 suggest that a substantial reduction in finished goods holding (a decrease in &f will slightly increase the likelihood that Fmished goods inventories wifl run short. An increase in the marginal desired finished goods inventory coefficient fil (column 4) means that firms are attempting to hold larger stocks, perhaps in order to avoid stockouts; but comparison of experiments no. 1 and no. 6 reveals that the variance of output and input bottbnecks worsen, perhaps because the increased sensitivity of inventory investment to changes in the fevef of output generates greater overait volatility in the output of the interactcting firms. And judging by a comparison of experiments no. 1, no. 7, no. 9 and no. 10, if firms attempt to keep their inventories closer to target levels (i.e. 6 is increased), the variance of total output increases and input bottlenecks become more of a problem. How production smoothj~g considerations enter into the picture is suggested by comparing experiments no. 12 and no. 12 with no. I: the larger the production smoothing coefficient the lower the variance of output, which might be expected. And as conjectured, when normal sales are more sluggish [i.e., the exponential smoothing parameter 0 of eq. (2) is small], the variance of output is reduced and stockouts are less common. Now consider the rn~han~~ by which firms manage stocks of ~~r~ha~ materials. A comparison of experiment no. 29 with no. 30 and experiment no. 31 with no. 32 suggests a seemingly contradictory picture: the more frequently management recalculates the (S,s) trigger points in an effort to align stocks with changing demand conditions the more frequently input bottlenecks are en~ou~tered~ Why is it that the effort to manage purchased materials more dose&, as indexed by the s~rnu~a~o~ parameter ~~~a~~j~ is not rewarded by a reduction in input botttenecks? Why is the business environment less stable (the variance of sales higher> when the fS,s) trigger points are frequently recalculated? Here is an example of the ‘fallacy of composition’: the greater the managerial effort individual firms allocate to ~ontro~~ng purchased materials the more serious the problem of cyclical instability and the incidence of stockouts! There is a second and equal& puzzling purchased materials anomaly. One would have thought that larger

M.C. Lovell, Simulating the inventory cycle

165

safety stocks would contribute to stability, which certainly makes sense from the perspective of the individual firm. But once again the fallacy of composition applies, for comparison of experiments no. 1 with no. 17 and no. 18 and no. 21 with no. 23 reveals that when firms attempt, acting individually, to avoid stockouts by adopting larger values of Smin (i.e. large safety stocks), precisely the opposite outcome is brought about; stockouts are more likely when S,in is large! How are these serendipitous simulation outcomes to be explained? These twin fallacies of composition arise because safety stock is related by (10) to the product of normal sales times Smin.” The impact on stock accumulation of rising sales during a cyclical upswing is magnified when Smin is large, and this heightened demand to augment safety stocks in the boom adds to the upward spiral of effective demand; and conversely, when the economy slips into a downturn, the drop-off in effective demand will be enhanced if firms respond by reducing the size of their safety stocks. However, the effect of changing sales on the size of the safety stock is blocked if firms seldom recalculate the (S,s) points. This explains why the output variance was low and input bottlenecks avoided, regardless of the magnitude of Smin, when firms did not recompute the (S,s) points - compare experiments no. 29 and no. 31. 3.3. Aggregate

simulations

How important is it to take explicitly into account the multi-sector multi-firm nature of the modern economy in analyzing questions of economic dynamics? Is one likely to be misled if one neglects problems of aggregation and approximates the economy by a single representative agent? Insight into the question of how accurately a single sector representative agent model can approximate a multi-sector economy with a multitude of economic agents is provided by table 3, which reports the results of running the same 32 experiments on an aggregate one-industry economy with but a single enterprise. The resulting model is similar to that analyzed by Lundberg and Metzler, but complicated by eon-negativity conditions and the variety of behavior modifications explained in section 2.*i It is not surprising to find that the aggregate model involving only a single firm utilizing its own output as its single input tells a quite different story about the incidence of input bottlenecks. In some respects, however, the “‘This effect might be less marked if the size of safety stocks were related to the square root of normal sales rather than by the proportional relationship modeled here. The nature of this link may depend on the extent to which cyclical expansions are met by the creation of new enterprises or plants, a factor that is excluded from the present version of the model. “The aggregate model utilizes a degenerate 1 x 1 input-output matrix. In an effort to establish comparability with the disaggregated simulations this parameter was set equal to 0.3878, the largest characteristic root of the 21 x 21 sector input-output matrix.

0.1

0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30 0.30

50.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0

100.0 800.0

200.0

z 0:1

0”:: 0.1 0.1

:::

:: 0:1

:i 0:1

30 31 32

18 17 19 20

0.10 0.10

0.20 0.10 0.10

0.00 0.10 0.10

0.00 0.00

0.00

0.30 0.30 0.00 0.00

2060 200.0 200.0 200.0 800.0 200.0 200.0 2,000.0 200.0 200.0 200.0 200.0 200.0 200.0 200.0

X:: 0.3 0.3

:: 0.3 0.3 0.3 0.3

8::

:3

8.: 0:3

::: 0.0 I.0

:::: 0.3

0.3

0.3

0.1 0.1

FI:f 0.1

::; 0.3 0.3 0.3 0.3

x::

Z:f

:f

0.1 0.1 0.1 0.1

Purchased mater&Is, simulaionsz

3 4 5 6 I 8 9 10 11 12 13 14 15 16

.-

@)--_- ?5)

F~is~~og~ sod simulal.ions:

No. MPC /Jo (1) (2) (3) -~ By;ngrk run: 200.0

Experimental conditions -Finished --I___~

z: 0:5

:I: 0.5

:: 0:5 0.5 0.5 0.5 0.5

0.5 0.5 0.5

f?; 0:8 0.5 0.5 0.5 0.5

X::

:I: 0.5

0.5 0.5 0.5

10.0

3.0 3.0 10.0

:::

10.0 10.0

t,: l:o 1.0

i:8

f.: 1:o

::I:

::“o

1.0

1::

:.: 1:o 1.0

::I:

:z 1:o

:::

;::

3.0 3.0 6.0 10.0 IO.0 20.0 20.0 IO.0

10.0

2.0

3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0 3.0

::: 0.0 1.0 0.0 1.0

8::

::: 0.5 0.5 0.5 0.5

0.5 0.5

0.5 0.0 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5

0.5 0.5 0.5 0.5 0.5

OS

OS 1.0

Recal) (9) --

P(Ss

r Prod Smooth Smin Smax (6) (7) -.-(8)

InputStocks ~_____

3.0

Table 3

:: 0:5 0.5 0.5 0.5

:: 0.5 0.5 0.5

:: 0:5 0.5

0.5

::;

::: 0.5 0.2 0.8

K

z::

8::

z:

0.5

@Expon S;;lpoth

Source sll$

x

8 14

::

:

8

:s’ 1

121 0

2:

:;

:!: 0

3:

:: 26 24

1:

15

Input Bottle (12) 78

FinGood Exhausted (13)

Experimental outcomes

One firm economy (PM plus FG simulations).

99.16% 6.98%

93.70% 6.99% 97.8 1% 95.17% 97.51% 99.55%

99.50% 97.67% 98.86% 97.55%

1,772 1,812 5,622 1,861 1,769 1,861

2,850 1,408 1,464 1,535 1,520

2,456

8,520 2,842 2,969 2,833 2,914 2,874 3,146 2,844 2,948 2,872 2,725 2,975 2,786 3,026 2,733

2,878

98.05%

88.4%

102.1%

93.6%

93.7% 97.9% 103.1% 94.9% 102.2%

102.2%

CT a-Output/ Output o-Sales (15) (16)

Ship/ Order (14)

G-32

G-58

G-58

Cycle Char (17)-

M.C. Lmell, ~~~ul~ting the inventory cycle

167

aggregate model is a surprisingly accurate approximation. It correctly identifies four out of the five disaggregate experiments reported on table 2 in which the multi-sector model encountered economic gridlock. The correlation matrix on table 4 indicates the degree of success with which the experimental results of the aggregate simulations approximate the multisector simulation outcomes.” Tentatively, it is reasonable to conclude that the negelect of ag~egation complications inherent in the use of representative agent macroeconomic models is less misleading than it had seemed reasonable to anticipate.

3.4. The production smoothing hypothesis Do firms smooth production? This is a long-standing conjecture, receiving attention from Holt, ~odigliani, Muth and Simon (1960) and from Beckmann (1961) and subjected to empirical test in early studies by Love11 (1959) and Fair (1971). Within the last decade the conjecture has been challenged by Blinder (1981, 1986), who argues that the observation that aggregate output has a larger variance than aggregate sales proves that manufactures do not use inventories to smooth production. In contrast to earlier studies, Blinder simply examined the overall ratio of the variance of output to the variance of sales in an effort to see if manufactures succeed in using inventories to dampen the impact of sales on output; he did not attempt through structural estimation to determine whether the desire to smooth production influenced the finished goods inventory and output scheduling decision. The controversy continues, focusing in part on questions of measurement error and the problems generated by the difficulty in finding data that have not been adjusted for seasonal variation.23 Insight into this controversy is provided by column 16 of table 2 reporting the ratio of the standard deviation of output to the standard deviation of sales (Goutput/~5sa,es). In the majority of the simulations Blinder’s ratio exceeds unity. It can certainly be said that in a majority of these experiments the simulated firms, on average, may have had more volatile output than shipments. But this does not mean that the firms were not trying to smooth production or that the variance of output would be less if this effort were not made. Blinder’s ~,,,~r~~/~,,i,, ratio is not a reliable indicator of whether firms are attempting to smooth production for it was highest for simulation no. 12, the simulation for which firms were making the strongest effort to smooth “There are 27 observations corresponding to the 27 experiments not resulting in grid-lock. %hali (1987) argues, using data for the cement industry, that firms do indeed smooth the impact of seasonal fluctuations in sales on output, a point that was missed by inv~tigators working with seasonally adjusted aggregate data. In a recent paper Fair (1989) updates his earlier study, arguing that the evidence still supports the production smoothing hypothesis.

InPutBn FGExha ShipOrd ir-Output crOut/uSa LnrJOutpu LncrOutaS Constrai *InPutBn *FgExha “ShipOrd *ooutput *crout/crs *LnuOutp *LncO/rS *Con&a

- 0.079 0.933 -0.956 0.875 - 0.029 0.748 -0.020 0.103 -0.217 - 0.221 0.195 0.084 0.449 0.022 0.428 - 0.247 SShift 1 1.000 0.000 - 0.005 -0.117 - 0.240 -0.191 -0.231 0.983 0.894 0.538 -0.683 - 0.049 - 0.208 -0.036 -0.201 0.788 InPutB 2 1.000 - 0.976 0.175 - 0.253 0.604 -0.251 0.183 -0.174 -0.292 0.224 0.051 0.230 0.007 0.205 -0.271 FgExha 3 1.000 - 0.803 0.174 -0.678 0.168 -0.188 0.173 0.272 -0.212 0.004 -0.342 0.072 -0.319 0.257 ShipOr 4

imo 0.113 0.928 0.121 0.038 -0.273 -0.321 0.290 0.502 0.530 0.368 0.512 -0.338 croutpu 5

1.000 0.392 0.999 - 0.268 -0.058 0.280 -0.249 0.152 0.744 0.170 0.760 0.146 uoutos 6 1.ooo 0.399 -0.064 - 0.280 -0.282 0.263 0.489 0.759 0.362 0.745 -0.318 LncOut 7

1.000 -0.258 -0.046 0.288 -0.254 0.149 0.155 0.163 0.771 0.157 LncrOaS 8

-0.635 - 0.036 -0.146 -0.033 -0.144 0.730 Constraint 9

0.848 0.482

l.OO@

Notes: Variables 1 through 6 correspond to the items in columns 11 through 16 on table 2. Variables 7 and 8 are the logs of variables 5 and 6. Variable 9 is the sum of the number of source shifts plus the number of input bottlenecks plus the number of times finished goods inventories were exhausted. Variables 10 through 17 (marked with an asterisk) were calculated with data generated with the single firm aggregate simulation experiments reported on table 3.

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

Correlation matrix 1 SShifts 1.000

Table 4 Comparison of multisector and aggregate simulations.

jj,

2 B X ?+ $ k

f 5’ ac

Fcl 9 f

ML.

Love&

Sindating

the inventory

cycle

169

prodLlction (z =0.8)!‘” ~urthe~ore, the ~~~~~~~/~~~~~ ratio does not accurately reveal whether the efforts of each firm to smooth production were successful in the aggregate. Camparison of the standard deviation of aggregate output (reported in column 15 of table 2) for simulations no. 1 (z =0.3), no. 11 (z =O.O) and no. 12 (z =0.8) demonstrates that, ceteris paribus, the fluctuations of aggregate output for the entire economy are indeed reduced when firms intensify their efforts to smooth productian. Froduction smoot~~g works, for it does indeed serve to reduce the variance of output; but Blinder’s G~“~~&,,,,, ratio is not a reliable index of how hard firms work at smoothing production or whether they succeed in attaining this objective.

How successfully do simple flexible accelerator models of inventory behavior explain the arti~~iai data generated by these simulation experiments? When the flexible accelerator model and its many variants have been applied to real world data the results can be appraised in terms of the reasonableness of the parameter estimates, goodness of fit, and predictive ability. Application of the same econometric procedures to artificially generated data will enable us to impose an additional check on the validity of estimation procedures because we can directly compare the estimated parameters with the actual parameters that generated the data. Also, we can ask how successfully the models cope with problems of aggregation and can look for any systematic errors encountered with the estimation technique. We shall look at both finished goods and purchased materials inventory. It turns out that many of the complaints made about the results obtained in applying these models to data for the U.S. economy apply also to the data generated experimentally by the hypothetical computer model. In particular, in the majority of the regressions the estimated speed of adjustment coe~cients are unreasonably small, fallin g substantially below the true parameter values.

Successive columns of table 5 present alternative estimates of the simplest flexible accelerator inventory model applied to the data generated by simulation no. 1. The regressions were all run with finished goods inventory

“‘The regression of oou,puJ5~~tes on the experimenta condition variables in columns 2 through 10 on table 2 yielded o&jr two significant percents, P(SsRecaf) and Expon~moot~ the production smoothing terms had a coeflicient of only 0.083, which was smaller than its standard error.

170

M.C. Love& Simulating the inventory cycle

?vi.C. Loveil, Simulating the inventory cycle

inv~s~~nt as the de~ndent variable. ” In terms of the simulation parameters, the basic equation that was tested takes the form:2”

N,-f-f,-, = S/3,+ ~~tS~les

-

6H, _

1

+

(Sales

-

171

model

Sides,_ 1)

This equation has several potential problems. First of all, it ignores questions of aggregation. Further, it relies on the change in sales as a proxy for the forecast error in order to allow for the possibility that businesses may systematically underestimate changes in sales.” Even so, table 5 presents results that an empirical researcher would regard as supportive. Admittedly, the least-squares regression reported in the first column is a disaster, for the coefficient on FGBEC, the holdings of finished goods inherited from the preceding period, has the wrong sign. The regressions in columns two and three, which add either the change in sales (a possible proxy for the forecast error) and the change in output (an indicator of production smoothing), are encouraging, although the ~urbin-Watson statistics provide cause for concern.‘* While regressions no. 4 and no. 5 attempt to cope with the problem of auto-correlation by including an autoregressive correction, the results are questionable because of the presence of the lagged dependent variable. The last column, estimated by Two Stage Least Squares, reports the preferred estimates. ” The estimated ii of only 0.259 is more than two standard deviations below the true value of 0.3 employed in generating the data; but using 6 =0.258 to cafculate estimates of the remaining parameters

rSAlternatively, adding sales to both sides of eq. (12) would yield an equivalent expression for output determination that has sometimes been used in empirica studies of the production decision. ‘@Ihe coefhcient of the change in output follows from eq. (41, which implies that the gap between smoothed versus planned output is f& -QP=r/(rl&$-Q,_l)~ 27Eerber (1953) conjectured on the basis of empirical evidence that firms systematically underestimate sales volume. An insignificant coefficient on the change in sales suggests either that firms may not sy~temat~~Ily underestimate sales or that there is time, within the observation period, to adjust production schedules to correct the systematic error. In applied work some investigators have tested for systematic change u~deres~~at~on or used survey derived estimates of sates expectations [e.g. Love11 fl959)] or have assumed that the forecast errors are random, as with Mills’ (1962) ‘implicit’ and Muth’s ‘rational’ expectations. 28As is now well known, the Durbin-Watson statistic is likely to be biased toward two because of the presence of lagged dependent variable H,_ 1. Durbin’s h statistic for regression no. 3 clearly indicates autocorrelation. 2PThis instrumental variable procedure, based on the contribution of Fair (1970), has been employed by Irvine (1981, 1988) and Fair (1989) in studies of inventory behavior. It was used here because of its simplicity, but in further work it would be useful to investigate its effectiveness relative to the many alternative ~timation procedures r~co~ended for handling the problem of lagged dependent variables when the stochastic disturbance is subject to autocorrelation.

172

M.C. Love& S~~ulat~n~ the ~nve~tQrycycle

from the other regression coe~cients yidds &= 199,4,” 8X=W2, and t= 0.496; these values
“%ince there are 4 x 21x84 firms, the estimated intercept for the representative firm is 433&&Mx 0.259)= 199.4. 31T~o of these exceptions, simulations no. 17 and no. 2, involved an exceptional number of input bottlenecks, which constitute a type of non-linearity that is not picked up by the simple econometric procedures considered here. The third exception, simulation no. 22, is not so easily explained. 32For simuiatian no. 1I, B-0; the TSLS coefficient on the change in oxSpat is four times its standard error but of wrong sign, which might lead the inyesti~at~r using a one-tailed test to correctly accept the null-hy~~hesis.

173

M.C. Louell, Simulating the inventory cycle

produced by the problem of aggregation rather limitation of the flexible accelerator model itself.

4.2. Stocks

ofpurchased

than

indicative

of a

materials

Can the flexible accelerator model also explain the behavior of stocks of purchased materials? There are obvious s~~~~~atio~ problems in applying this model to data generated by (S,s) stocking practices: First of all, the link between stocks and output (or orders), at least at the level of the individual firm, is unlikely ta be positive for short periods of observation unless firms revise the (S,s) trigger points with considerable frequency; but the flexible accelerator class of inventory models does not explicitly incorporate the probability that firms will recalculate the (3,s) trigger points. Second, for stocks of individual inputs, although perhaps not for the aggregates, one might expect a square root rather than linear relationship between the reorder quantity (S,s) and output. 33 While these limitations are obvious, it is not clear how these problems are best addressed in analyzing aggregate data.34 Therefore, it is of interest to examine the degree of success achieved by the Bexible accelerator model in explaining the aggregate data on investment in stocks of purchased materials, although it may be too much to hope that the problems with the model at the level of the individual economic agent may wash out when looking at the aggregates. A variety of attempts at explaining the behavior of the purchased material data generated by simulation no. t are presented on table 6. There are an impressive number of significant coefficients and the Durbin-Watson statistics are satisfying. While the flexible accelerator model may be more successful at explaining the aggregates than we have any right to expect, there are problems: Although much tighter fits are obtained when both output and orders are included in the reiatianship (regressions no. I and no. 7), one might have anticipated that a high level of orders might at best have a positive impact on stocks but that an increase in output, given orders, should tend to reduce stocks of purchased materials, at least pending the a3At the industry level of aggregation there might be a square-root relationship in the longer run if adjustment involves an increase in the individual output levels by a stable number of &ms, but the relationship might be finear if secular growth is satisfied by an increase in the number of asfabfishments. Thus the form of the relationship depends on a host of industrial organizational issues. For the simulations used in generating the artificial data analyzed in this paper the number of firms was frozen, which suggests that the square-root relationship should dominate. 34Bhnder (1981), under the assumption fhat ah firms have the same (S,s) trigger points, obtains as a iinear approximation an equation that is similar to the flexibte accelerafor but nonlinear in the parameters. In future research it would be of interest to determine how we11 Hinder’s a~~roxirnati~~ works when all Erms do not have the same (22,s) points, as in the simulation model used in this paper.

J.E.BO.-C

0.668 96 44.117 2.314

R-square-adj: Number of observations: Standard error adj: Durbin-Watson stat: 0.613 96 48.259 2.345

2 LS

Regression coefficients and standard errors: C - 23.509 - 170.468*** (63.264) (55.854) ORDERS -0.239*** (0.059) 0.372”** 0.156*** OUTPUT (0.055) (0.013) -0.124*** -0.101*** PMBEG (0.016) (0.017) OUtpUtsxP 1.0 1.0

1 LS

Equation number Estimation method: Iterations to converge:

Dependent variable: Investment in purchased material stocks

Simulation # 1: Smin=3; Smax= 1, P(SsRecal)=0.5

4.449 (35.251) -0.102*** (0.017) 0.640 (0.820)

- 446.521 (1,001.842)

0.614 96 48.188 2.374

3 NLS

l/2

17.783*** (1.450) -0.101*** (0.016)

-612.577*** (82.127)

0.620 96 41.823 2.386

4 LS 5

17.794 (146.307) -0.101*** (0.017) 0.500 (0.815)

- 672.576 (1,639.244)

0.616 96 48.082 2.386

NLS 1

-0.170 (0.102)

0.156*** (0.011) -0.097*** (0.014) 1.0

183.227*** (47.567)

0.620 96 47.819 2.003

6 TSLS 1

-0.157 (0.103)

-42.978 (57.514) -0.227*** (0.058) 0.361*** (0.053) -0.119*** (0.015) 1.0

0.672 96 44.426 1.995

7 TSLS 3

Notes: The OutputEXP row reports the exponents of the output variable, which was constrained to equal unity in regressions 1, 2, 6 and 7 and to l/2 in regression 4. The exponent was estimated to be 0.640 when unity was used as the starting value - but the procedure failed to converge. It was estimated to be precisely 0.5 when l/2 was used as the starting value. Standard errors in parentheses. ** indicates 2 < 1t / < 3; *** inducates 3 < 1t 1.Eq. (3): Convergence not achieved after 100 iterations. Eq. (6): Instrument list: C ORDERS (- 1) OUTPUT OUTPIJT (- l)PMBEG (- 1). Eq. (7): Instrument list: C ORDERS ORDERS (- 1) OUTPUT OUTPUT (- 1) PMBEG (- 1).

AR(l)

Table 6 Purchased materials regression results (4 firms in 21 industries).

ML. Love& Simulating the inventory cycle

175

ftf tke (S,sf points; these sign expectations are not satisfied. And as with empirical studies of U.S. data, the estimated speeds of adjustment (the coefficients of PMBEG) appear to be unacceptably small. Furthermore, while the use of the square-root of output (regression no. 4) fits slightly better than the linear relationship (regression no. 2), the difference is negligible, Two alternative nor&near estimates are presented of the exponent of output; regression (3) used an exponent of one as the starting value for the iterative calculations while no. 5 used the square-root exponent of i/2 as the starting value. The resulting estimates are far from satisfactory, in part because the non-linear estimation procedure had not fully converged after 100 iterations;35 the reported estimates suggest that the exponent of sales may not be signi~cantly different from zero! The flexible accelerator mode1 did rather better than we have any right to expect when called upon to explain aggregate data on investment in purchased materials generated by the 27 simulations that were not terminated prematurely by economic gridlock. Suppose an investigator regards the evidence as supportive if the coefficient on lagged stocks is sig~~~antly less than zero and greater than minus one and, in addition, the sum of the orders plus the sales regression coefficients is greater than zero. Then our investigator would have concluded fram the aggregate data that the flexible accelerator model was supported in 18 of the 27 ex~riments. Looking at the data for a single firm, the investigator would have concluded that the evidence was favorable in 16 out of 27 cases. Unfavorable estimates were generated for the three ~mula~ons for which ~(~eGal~)=O with both the aggregate and the firm data, which is obviously appropriate in that the (S,s) points are never recalculated. And it is not surprising to find that the flexible accelerator model also runs into trouble with both aggregate and disaggregate data in experiments no. 13 and no. 15, which involve the use of highly smoothed exponential estimates of normal sales, In sum, an empiricist investi~aiing data generated by f&s) stocking rules would appropriately reject the flexible accelerator model if firms seldom or never recalculate the (S,s) points in response to fluctuations in sales. But in the great majority of the other simulations, the results appear supportive, and this is so with data at the firm level as well as with aggregate data. r~al~ula~i~~

35With the default conversion criterion value of 0.0135, version 6.5 of the popular TSP Econometric Program reported convergence for the two alternative starting values of the nonlinear estimation problem, but two quite different parameter estimates were generated! The clear moral of this unhappy experience is that the results obtained with packaged non-linear estimation procedures should not be interpreted at face value but must be checked with dternative prehminary startup values of the parameters. In an effort to cope with the probIem of divergent estimates, the ~nvergen~ criterion was reduced to OBtll and the max~um number of iterations was increased to 100. Even so, the TSP program reported convergence to divergent estimates for four of the 27 simulation sets.

176

M.C. Love& Simulating the inventory cycle

5. Conclusions

The investigation of artificial data generated with a multi-sector inventory cycle model supports the following conclusions, some of which may be more provocative than others: 1. The simulation results are consistent with two major conjectures suggested by previous theoretical research on the stability properties of multi-sector inventory cycle models derived without regard for such complications as non-negativity constraints on output and stocks: (1) The system is less likely to be stable if either the marginal desired finished goods inventory coefficient or the marginal propensity to consume increase. (2) Instability is more likely when the behavior of enterprises is less subject to inertia, firms moving more promptly to adjust finished goods inventories to current economic conditions, subjecting sales to less exponential smoothing in calculating normal sales, and relaxing their efforts at production smoothing. But there were surprises. In particular, it was surprising to find that when individual firms carried larger stocks of purchased materials inventory, relative to output, the system was less stable, stock-outs more frequent, and gridlock more likely. It was explained that this anomaly arises because the re-setting of the (S,s) trigger points in response to persisting changes in sales volume causes more volatile investment in purchased materials inventories over the course of the cycle when firms strive to maintain a high safety-stock to sales ratio. 2. Many of the dynamic properties of the multi-sector model involving a host of interacting economic agents are fairly well approximated by a one-firm one-industry aggregative model. 3. The ratio of the variance of sales to the variance of output turns out to be an unreliable indicator of how seriously firms attempt to smooth production or whether such efforts, in the aggregate, are serving to iron out economic fluctuations. 4. For the most part, the flexibile accelerator model provides a rather better framework for econometric research than we have any right to expect when studying aggregate finished goods inventory data. However, the application of standard econometric procedures to the artificial data does yield extremely low estimates of 6, the speed of adjustment. This investigation suggests that the ‘too slow’ estimates reported in many econometric studies of United States data may result at least in part from an inherent estimation bias in fitting the model to aggregate data rather than from a limitation of the flexible accelerator model itself. 5. While the flexible accelerator model is a most imprecise specification of (S,s) behavior, parameter estimates obtained by applying this model to both firm and aggregate simulation data appeared to be acceptable in the

M.C. Love& Simulating the inventory cycle

171

majority of cases; not surprisingly, the model did not work when the (S,s) points were seldom or never recalculated. 6. P(Recalc), the frequency with which firms recalculate the (KS) trigger points, is a parameter that has been generally neglected in discussions of the inventory cycle, econometric studies of inventory behavior, and the management science literature. The simulations show that this neglected parameter may be of critical importance in determining the stability of the economy. Frequent recalculation of the (S,s) trigger points may lead to economic instability and contribute to increased stockouts and production bottlenecks. However, if firms rarely recalibrate the (3,s) benchmarks, changes in carrying costs, such as might result from a shift in monetary policy, will not affect inventory accumulation. This type of inertia may partially explain why empirical investigations of inventory behavior usually fail to uncover any effect of interest rate changes on inventory investment.

Appendix A: Excerpt from Pascal program The heart of the 990 line Pascal program is the 15 line procedure generates each firm’s output decision.

that

Procedure FirmOutputDecision(var firmtf: FirmT); {Sets output on basis of projected demand using purchased materials on hand, subject to availability~ Begin With Firmtf do begin {Process firm f of industry t} No~alOrders: = Theta*Orders + (1.0 - Theta)*NormalOrdersLag; NormalFinGoodInv: = Beta0 -I-Beta1 *NormalOrders; TargetFinGoodInv: = Delta*NormalFinGoodInv + (1.0 - Delta)*FinGoodInvLag; (Eq. (3)) PlannedOutput: = Orders + TargetFinGoodInv - FinGoodInvLag; NonNeg(PlannedOutput); {Output cannot be negative) SmoothedOutput: =( l.O-Tau)*PlannedOutput + Tau*OutputLag; (Eq. (4)) NonNeg(SmoothedOutput); Output: = SmoothedOutput; End; End; End; (FirmOutputDecision}

178

M.C. Looell, Simulating the inventory cycle

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