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9-1970

Production, Price, and Inventory TheoryGeorge A. HayCornell Law School, george.hay@cornell.edu

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Recommended CitationHay, George A., "Production, Price, and Inventory Theory" (1970). Cornell Law Faculty Publications. Paper 1154.http://scholarship.law.cornell.edu/facpub/1154

Production, Price, and Inventory TheoryBy GEORGE A. HAY*

This paper is an attempt to derive em-pirically testable hypotheses regarding theprincipal determinants of firms' decisionson production, price, and finished goodsinventory. The general approach to theproblem is that many of the same factorswhich affect the optimal value for one vari-able will also influence decisions on theother two, and that a "proper" modelmust take into account the interdepen-dence of these variables and the simul-taneous nature of the decisions involvingthem. This is in contrast to literature onthe theory of inventories (see Paul Darlingand Michael Lovell) in which the firm isassumed to determine the optimal level ofinventories with the rate of productiontaken as given and with price held con-stant. Similarly there are theories of priceformation (see Otto Eckstein and GaryFromm) in which the rate of production isassumed to have been determined pre-viously, and in which the level of inven-tories is often ignored entirely. The pres-ent paper will attempt to present an "in-tegrated" model of firm behavior in whichdecisions on all relevant variables areassumed to result from a single optimiza-tion process.

A principal distinction between thisstudy and previous work in this area (seeGerald Childs and Charles Holt) is the in-clusion of price as one of the decision vari-ables.^ In the past the assumption has

* Assistant professor of economics at Yale Univer-sity. I wish to acknowledge the valuable advice ofGerald Childs at an early stage in the preparation ofthis paper. A referee also provided useful suggestions.

' Edwin Mills developed a model which includedprice. He was able to derive approximations to the truedecision rules which held up reasonably well underempirical investigation. The present study attempts toderive an exact set of decision rules.

been made that price is fixed and thereforethat quantity demanded is a datum to thefirm. In anything but a purely competitivemodel, however, the firm does exercisesome control over price. The rational firmwould use its current pricing policy to se-lect the specific price-quantity combina-tion on the demand curve that best con-tributes to its overall scheme of profitmaximization. Thus the firm whose de-mand curve is not constant over time butfluctuates from period to period on a ran-dom and/or seasonal basis might viewprice adjustments as one means of achiev-ing production stabilization. If this wereso we might expect to find that an increasein demand would be met by continuing toproduce at or near the previous rate andraising price to clear the market. Morerealistically, the entire kit of adjustmenttools—inventory, backlog, and price—would be used in some combination toabsorb all or part of the increased demand,the specific result depending not only onthe particular cost structure assumed, butalso on the firm's estimates of what de-mand will be for several periods hence.The important point is that price mustcertainly be included as one of these tools.

In the remainder of the paper we con-struct a model which includes many of thevariables which are important at the indi-vidual firm level, and which treats deci-sions regarding those variables as beingessentially interdependent. The subse-quent section discusses the behavioralassumptions which underlie the model andexpresses the model in mathematical form.The first-order conditions for maximizingexpected profits generate a set of lineardecision rules for production, price, andfinished goods inventory. On the assump-

531

532 THE AMERICAN ECONOMIC REVIEW

tion that firms do attempt to maximizeexpected profits, these decision rules aresuitable for empirical investigation withthe appropriate data. In Section II, themodel is solved numerically with represen-tative cost parameters. The resulting de-cision rule coefficients provide predictionsregarding the actual regression equationswhich should be fulfilled if the model ac-curately reflects the working of the realworld. Finally, in Section III, the regres-sions are performed on two industrygroups, and the results compared with thepredictions.

I. The ModelBehavioral Assumptions

The model is intended to represent afirm which chooses the levels of the vari-ables it controls in order to maximize theexpected value of discounted future profitsover a time horizon. The variables in-volved are the rates of production andshipments, the level of finished goods in-ventories, the backlog of unfilled orders,and price. These variables are not inde-pendent, being related by various defini-tional and market constraints, includingthe demand equation. Each of the vari-iables serves a particular function, andassociated with each of these variables arecertain costs. These functions and costsform the basic elements of the model.

A positive level of unfilled orders is,practically speaking, an unavoidable phe-nomenon for most firms which undertakeany production to order. The timing of thearrival of new orders is not in general sub-ject to control by the firm. The develop-ment, design, and production of each ordertakes time, so that there will typically besome work still in process when a neworder is received. Even beyond this, how-ever, a higher level of unfilled orders maybe a useful alternative for the firm becauseit permits smoothing of production withinthe period and accumulation of optimal

size production batches. This is particu-larly true when production is a multi-stage process and several items which areeventually individualized to the specificrequirements of their respective purchaserscould nonetheless go through several stagesof the production process together. On theother hand, there are costs associated witha high level of unfilled orders. As the back-log gets larger and the lead time longer,there is increased danger of cancellation oforders, penalty costs for expediting par-ticular orders, and probable loss of futuresales.

This suggests that there is some positivelevel of unfilled orders which balances thecost savings attributable to an order back-log and the penalties associated with toolarge a backlog so that the net saving ismaximized. We might refer to this as the"desired" level of unfilled orders in thesense that, if there were no other consid-erations involved, this is the level the firmwould try to maintain. We will assumethat the desired level of unfilled orders, U*,is a linear function of the rate of produc-tion:

As stated by Childs:

Penalty costs for cancellation oforders, for expediting particular ordersin response to customer requests andthe probability of loss of future salesall increase as the size of the backlogincreases and the lead time lengthens.However, more flexibility in produc-tion arises as backlog mounts. There-fore, as backlog and lead time increasethe costs related to inflexibility of pro-duction decline. Average lead time isapproximately the ratio of backlog tothe rate of production. Then for everyrate of prodtiction, the cost associatedwith varying size of backlog is the sumof monotonically rising and mono-tonically falling components over therelevant range and has a minimum, U*,the optimal level of U^.. [p. lO]̂

»It is probably true that f/t* is a function of only

HAY: PRODUCTION, PRICE, AND INVENTORIES 533

Furthermore, we will assume that thecost of deviating from the desired levelincreases quadratically with the size of thedeviation and is the same in either direc-tion. Thus the net contribution to totalcost of a given level of unfilled orders isgiven by:

Cn + ci(t/t - UfY

where Ut is the actual level of unfilledorders at the end of t, and Cn is a constant(which may be negative).

Thus we have separated all the costsand cost savings specifically associatedwith an order backlog into two parts; thefirst being the net contribution to totalcosts of the desired level, Cn, and theremainder reflecting the additional costsof deviating from the desired level, recog-nizing that when all costs which the firmincurs are considered, the optimizing levelof unfilled orders may differ from the de-sired level. The rational firm may, forexample, be willing to deviate slightly fromits desired level if doing so will make itpossible to avoid a substantial increase (ordecrease) in the rate of production fromone period to the next.

A similar argument can be used to ex-plain the existence of a positive level ofinventories. Certain cost savings accruefrom the added flexibility made possible inproduction; on the other hand, an inade-quate level of inventories can have seriousconsequences for future sales through lossof goodwill which may be caused by a"stock out"—a firm's inability to fill anorder from a customer who requires im-

tliat part of Xt which is production-to-order. However,introducing production-to-order and production-to-stock as separate variables complicates the analysis andfurthermore requires an arbitrary aggregation pro-cedure at the end since the only observed variable istotal production. In addition, the nature of the resultsnot is affected so long as the cost of changing the rate ofproduction (discussed below) is independent of the mixbetween order and stock. (Similar remarks are ap-plicable to the inventory decision.)

mediate delivery. A firm not only loses outon the sales corresponding to that particu-lar order, but may suffer the loss of futuresales as well if the disappointed firmswitches preferences in favor of anothersupplier. This effect might most conven-iently be handled as a cost whose expectedvalue is associated with an inadequatelevel of finished goods inventories.

We assume that the desired level of in-ventories is related to the quantity to beshipped during the period. This can beconsidered an approximation to the opti-mal lot size formulas in the operations re-search literature (see Holt et al. pp. 56-57). Specifically we assume that the rela-tion:

St = C23 + C245t

is valid over the relevant range, where H*is the desired level of finished goods inven-tories at the end of the period t, St is ship-ments, and C23 and Ca are appropriate con-stants. Furthermore, we assume that thecost associated with being away from thedesired level of inventories may be ap-proximated by a quadratic such that theoverall contribution of inventories to totalcosts is represented by:

C21 + ciiH, - H^Y

where C21 reflects the net effect of the "de-sired" level of finished goods inventories.

Unit costs of production within a timeperiod are assumed constant. Beyond this,however, are costs associated with changesin the rate of production from period toperiod which are independent of the actualievel of production. These include varioussetting up costs and costs of hiring and fir-ing where a change in the work force is re-quired. We express these costs simply as:

This captures the notion that changesare costly in either direction and that largechanges are likely to be relatively more

534 THE AMERICAN ECONOMIC REVIEW

costly over a certain range than small ones.The general idea is that firms do seem toattach considerable weight to the stabilityof the rate of production and the size of thework force. This has the effect of makingthe previous period's output a factor ofconsiderable importance in the determina-tion of the output for the current period.

Demand for the product of the firm (inthe sense of the entire demand curve) isnot constant over time but is assumed toshift from period to period, in responsepartly to random factors, and partly tofactors which the firm observes but cannotcontrol (e.g., the level of national income).To make the model amenable to analyticaltreatment we must give a specific form forthe demand curve. In this model, there-fore, we assume that demand is of theform:

Ot = Qt - bPt

when Ot is new orders and Pt is price inperiod t. This is a linear demand curve ofconstant slope b which shifts over time inparallel movements, the extent of the shiftdetermined by the quantity interceptterm Qt. For reasons discussed below, thisdemand curve is the schedule of the quan-tity demanded from the firm at variousprices when all firms charge the same price.

This leads to the final cost to be con-sidered, viz., that associated with changesin price. It may not be common to thinkabout costs of changing price yet we ob-serve that firms are generally reluctantto do so in terms both of frequency andmagnitude. There are, of course, certainout-of-pocket costs which must be met—the necessity of revising price books andpossibly some additional advertising ex-pense to announce the new price.

Probably a more important influence onfirm behavior, however, are those costswhich are never actually observed but maybe thought of more as a type of oppor-tunity cost, the amount of money a firm

would be willing to pay, ceteris paribus, toavoid changing price. This reluctance isdue generally to the risk associated withimperfect knowledge about reaction bycompetitors to a price change. We citeWilliam FeUner for a description of thereasoning involved:

Each firm knows that others havedifferent appraisals [of the appropriatepolicy for the maximization of industryprofits] and that they are mutuallyignorant of what precisely the rival'sappraisal is. Consequently, no firm canbe sure whether the move of a rival istoward a profit-maximizing quasi-agree-ment or towards aggressive competi-tion; and no firm can be sure how itsown move will be interpreted. This iswhere the desire to avoid aggressivecompetition enters as a qualifying fac-tor. Even where leadership exists, theleader's moves may be misinterpreted asaiming at a change in relative positionsrather than as being undertaken in ac-cordance with the quasi-agreement.fp. 179]

There are, of course, other argumentswhich might be offered for the inclusion inthe criterion function of a penalty for pricechanges. The important consideration isthat the firm acts "as if" there were costsattached to changes in price, whether theseare explicit, out-of-pocket costs, or moreof an opportunity-cost, implicit type whicharises from uncertainty about rivals' reac-tions to its own price decisions.

We might assume initially that the costof changing price can be expressed as:

The symmetry assumption regarding thecost of changing price is perhaps bother-some but extremely convenient mathe-matically. If we regard the demand curveas based on the assumption that otherfirms will always imitate price changes byour firm, then the quadratic penalty canbe taken to represent the fear that for aprice increase, the firm will not be followed,

HAY: PRODUCTION, PRICE, AND INVENTORIES 535

and on a price decrease, the firm will beundercut.

There may be certain circumstances,however, under which the reluctance tochange price might be considerably dimin-ished. If the incentive for a firm to initiatea price increase is the result of an increasein labor or raw materials cost (which pre-sumably would affect all firms in the in-dustry to a similar, if not identical, de-gree) , it is likely that such a move would bewelcomed by rivals as an opportunity forthem to restore the margin which existedprior to the increase in direct costs, andhence the price increase would be matched.Indeed, failure to do so by any single firmmight well be interpreted as aggressivebehavior by rivals in the same sense as aprice cut in the absence of any changes incost. To the extent, then, that firms in theindustry foUow this type of increase com-pletely, the cost associated with initiatingsuch a change is likely to be insignificant.(A similar argument can be given for pricecuts which are occasioned by a drop indirect costs.)

Therefore, we can amend the model toassume that the firm is sensitive to changesin markup rather than changes in the ab-solute level of price. Cost-induced pricechanges can be initiated without fear of notbeing followed and hence the firm does notattribute a penalty to such a move.Changes in price which are not related todirect costs, or failure to change price whendirect costs vary will be assigned a cost inthe firm's decision-making process. In theterms of the model we are replacing

with:

c,[{Pt - Vt) - (Pt-i - V^i)]'

where Vt represents the direct unit costsof production (labor, capital rental, andraw materials) in period t.'

• It may be that a more complex lag structure would

Derivation of Linear Decision Rules

We can now bring the cost and revenueterms together into a single equation sothat the conditions for optimization can bederived. It is perhaps best to begin bysummarizing the notation. Let:

Zt=rate of production in period tPt=price in period tt/t=level of unfilled orders (backlog)

at the end of period tt/t*=desired level of unfilled orders at

the end of period ti7,=level of finished goods inventories

at the end of period tfft*= desired level of finished goods in-

ventories at the end of period tOt=new orders in period t5t= shipments in period t

These variables are constrained by thefollowing identities:

The demand equation relates the endog-enous variable Ot to the decision variablePt and the exogenous term Qt:

Ot = Qt - bPt

We will treat the problem initially asone of certainty. Furthermore we assumethat the firm discounts future profits ac-cording to a discount factor X(O<X<1).Therefore, to maximize discounted futureprofits over an iV-period horizon, we maxi-mize the Lagrangian expression in equa-tion (A) shown on the following pagewhere St and 7t are Lagrangian multipliers,and X a discount factor. After taking deri-vatives for any period t with respect to thedecision variables and allowing t to take on

be appropriate according to the speed with which in-creases in direct costs are recognized and transmittedinto prices, but in the absence of any detailed empiricalknowledge on this process and in the desire to use asingle model for many industries, this structure will beretained.

536

(A)

L

THE AMERICAN ECONOMIC REVIEW

= '^\ <PtQt — bPt — Cu — ci{Ut — Cu — cuXt) — C21 — Ci{Ht — C23 — CiStt=i I

- Xt_i)' - C4[(Pt - Ft) - (Pt-x - F^i)

all values to iV(where TV is large), the sys-tem is solved according to the Z transformprocedure outlined in Holt et al.* The endresult is a set of linear decision rules forproduction, price, and finished goods in-ventory of the form:

(1) Zt= ^ 11X

(2)

(3) Ht=An

where the ^'s are constants. There are alsorules for Ut and Sx which can be derivedfrom the first three.

Since Qt, Qt+i. . . and Ft, Ft+i. . . arenot known quantities but random vari-ables, the firm can do no better than maxi-mize the expected value of profits. Accord-ing to the Simon-Theil theorem, the samedecision rules apply provided we substitutefor Q and V their expected values, Q andV'. These equations are used to determinethe first period's decision for X, P, and H.Each period the rule is used again, withthe current period being treated as thefirst.

In a normative model, the firm would

* The exact solution is not presented here due to lackof space.

make a forecast of future Q and F andinsert the resulting values into its decisionrules. Since we do not have any informa-tion about what forecasts the firms used,for purposes of regression analysis we mustguess at the specific set of forecasts onwhich the decisions were based. Followingprevious studies of this type by Childs andHolt et al., we have used Q,t=Qt, i.e., onthe assumption that the firm is on averagean accurate predictor of future events, thedemand that actually comes about is usedas a proxy for what the firm had antici-pated at the time the decisionswere made.̂ '®The same formulation is used for V.

Aggregation

The model outlined above and the re-sulting equations for production, price,and finished goods inventory dealt withthe decision process of the individual firm.The available data cover the industryaggregates corresponding to these vari-ables. Problems of aggregation have beentreated elsewhere (see Theil) and will notbe discussed here. The main point is thatthe coefficients of the industry "decision

' Naive and distributed lag expectations were triedin our regressions and did not do as well as the perfectforecasts.

• Qi is not directly observed but can be obtained forpurposes of regression analysis by the inverse relationQt=Ot,-\-hPt. Although b is not known either, an apriori "reasonable" value (see Section II) can be used.As a check, alternative values of b were used to generatethe Q series with very little effect on the regressionresults. Note that for the firm Q is a true exogenousvariable. The use of {O-\-bP) as a proxy for what theeconometrician cannot observe does not change thereduced-form nature of the actual decision rules.

HAY: PRODUCTION, PRICE, AND INVENTORIES 537

rule" wiU not in general be a simple func-tion only of the corresponding micro-coefiicients. To achieve an unbiased set ofrestrictions would require knowledge atthe individual firm level which we simplydo not possess. In the absence of suchknowledge the sample calculations dis-cussed below will assume that the industrycan in fact be treated as a single large firmand will assume that any restrictions onthe micro-coefficients will carry over, withappropriate scaling, to the correspondingmacro-coefficients.

II. Calculation of Decision Rulesfor Particular Cost Structures

This section reports on an attempt toestablish some of the qualitative proper-ties of the system which has equations(l)-(3) as its reduced form. Specifically,some estimate should be provided as tothe range within which the regression co-efficients are likely to lie if the observeddecisions are made in a manner approxi-mated by the decision model. Further-more, since the An in the reduced formare functions of all the parameters in theoriginal structure {b, X, and the c's), it isdesirable to know how the ^ ' s will be af-fected by different assumptions about thevalues of these parameters.

Difierent firms or industries to whichthe model might be applied have differ-ent cost parameters, and one wants toknow how this fact will affect the [^'s].In addition, it is known that within afirm or industry these parameters donot usually remain constant over theperiod covered by a time-series sample.Thus, if the regression coefficients arevery sensitive to changes in the under-lying parameters, results will be poorwhen the regressions are estimated fromempirical data even if the basic decisionmodel is correct. [Mills, p. 139]

For a less complex structure, the coeffi-cients of the reduced form equations couldbe derived analytically as functions of the

parameters of that structure. With thepresent model, however, that approachdoes not appear to be feasible. It wouldrequire, in addition to matrix manipula-tion and determinant evaluation, theanalytic solution of an eighth degree equa-tion.

Given the infeasibility of an analyticsolution, an alternative approach is at-tempted in the present study. The systemis solved for specific values of the cost andrevenue parameters. As these parametervalues are changed in successive trials, theresults are examined to determine the sen-sitivity of the A ij to the specific parametersused. Hopefully, for a wide range of valuesfor the b, X, and c's, the A coefficients willretain at least the same signs, and to adegree, the same relative magnitudes. Theextent to which the coefficients are sensi-tive to the parameters gives some indica-tion of the results which might be expectedwhen parameter values other than the onesused in the samples are appropriate.

Parameter SpecificationsThe first stage of the experiment con-

sists of the selection of an initial set of costand revenue parameters to be used to ar-rive at a specific numeric solution for thederivation of the decision rule coefficients.Including X, the discount factor, and b, theslope of the demand curve, there are twelveparameters to be specified. The followingcharacteristics of the data will help to setthe order of magnitude for these param-eters.

Price is in index form and is of the orderof 100, with a standard deviation of ap-proximately 3.8. The physical series (ship-ments, inventories, etc.) are reported inmillions of dollars. Shipments, production,and new orders averaged approximately$1000 million with a standard deviation ofapproximately $165 million. Inventoriesand unfilled orders each average around$600 million with a standard deviation of

538 THE AMERICAN ECONOMIC REVIEW

IndependentVariable

DependentVariable

XtPtBt

TABLE

.260- . 1 6 9

.195

1-DECISION

Pt-i B

- . 0 2 8 - ..903 - .

- . 0 0 7

RULE COEFFICIENTS FOR

345058354

. . .

.495

.066- . 0 0 1

Qi

.492

.156- . 0 0 1

A PARTICULAR

Ot+i

.089

.106

.024

.019

.081

.012

COST STRUCTURE

. . .

.028- . 9 0 3

.007

Vt

- .0321.028

- .009

. . .

- .004- .010- .003

Vt..

- .001- .008- .001

" The two infinite series were truncated at this point. This is justified by the exponentially declining weights in theexpansion of Q(Z) and V(Z).

approximately $100 million and $80 mil-lion, respectively.'

With these figures in mind, the followingwere assumed to be the values of the costparameters:

i) Cn, Cn, Cn, and C23 drop out entirelyor affect only the intercept term and wereassumed equal to zero.

ii) b=.O5. Given the magnitude of theprice and new orders data, this correspondsto a price elasticity of .5 which seems rea-sonable given the assumptions about thedemand curve (i.e. that it specifies quan-tity demanded from the firm when allfirms charge the same price).

iii) X=.99. Since the decision period isone month, this corresponds to an implicitannual discount of approximately 12 per-cent.

iv) ci4=C24= .6. Thus it is assumed thatthe observed long-run ratios betweeninventories and shipments and unfilledorders and production approximate thedesired relationship.

v) Ci=150; C2=100; C3 = 35; C4=5. Itwas assumed that if the actual level of anyof the variables deviated from the desired(or previous period's) level by as much asa standard deviation (approximately), thecost would equal 10 percent of total rev-enue for that month.

The results of the initial sample calcu-

' There are some small differences between the twoindustries studied for all of these figures, but notenough to warrant separate treatment.

lation are presented in Table 1 above. InTables 2-a, b, and c we attempt to providesome notion of how the A ij depend on thestructural parameters. In each of the eighttrials a single parameter, indicated in theleft-hand column, was varied from theinitial set of values. In the case of X, thevalue was lowered to .89. In all other cases,the parameter under consideration wasdoubled.

Looking ahead to the regression analy-sis for the production and price equationsthe results of the sample calculations aregenerally encouraging. The algebraic signsfor all the coefficients remain unchangedin every trial. Furthermore, the magni-tudes of many of the coefficients were rela-tively insensitive to changes in the param-eters indicating that time-series analysismay yield meaningful results, even thoughthe cost and revenue structure of the firmsinvolved have undoubtedly undergonesome degree of change over the time per-iod studied. For the inventory equation,however, the sample calculations produceambiguous results, with only the signs ofXt_i and Ht-i remaining the same inevery trial.*

It is interesting to note that the produc-tion and inventory rules are generally un-affected by different specifications regard-ing the cost of changing price, d, and the

' This is only partially shown in Table 2. Many otherparameter specifications were tried but are not reportedhere.

HAY: PRODUCTION, PRICE, AND INVENTORIES 539

slope of the demand curve, b. Similarly,the rule for price is relatively insensitiveto Cu and Cu- As evidence of the interde-pendence among decisions, however, wenote the strong impact of Cz, the cost of

changing the production rate, on the pricedecision rule.

In terms of individual coefficients it isworth noting that the coefi&cients of Ut~iand Qt are virtually identical in the pro-

TABLE 2-a: PRODUCTION DEasiON RULE .X t̂: CHANGE FROM INITIAL SET UNDER ALTERNATIVE COST STRUCTURES

Value ofParameter

which ischanged

X=.89e, = 300Cs=200c,= 70C4= 10

6=.1O

Xt.i

+ .003- .053- .021+ .096- .003- .111- .011+ .015

Pt-i

- . 0 0 1- . 0 0 1- . 0 0 1+ .004- . 0 0 1+ .006- . 0 0 5- . 0 2 3

Bt-i

0- . 0 2 6- . 0 2 8+ .078- .001+ .033+ .036+ .007

Ut-i

+ .005+ .053+ .046- . 1 3 2+ .001- . 0 2 1+ .090- .008

Qt

+ .005+ .053+ .046- . 1 3 2+ .002- . 0 2 0+ .090- . 0 1 0

Q.+i

- . 0 0 4- . 0 2 1+ .012- . 0 0 6+ .002- . 0 3 1+ .015- . 0 0 5

Qt+>

- . 0 0 1- . 0 0 6+ .002+ .003+ .002- . 0 0 5- . 0 0 2- . 0 0 3

Ft-i

+ .001+ .001+ .001- . 0 0 4+ .001- . 0 0 6+ .005+ .023

F.

+ .001- . 0 0 1

0+ .007+ .001+ .006- . 0 0 4- . 0 2 5

F.+.

00

- . 0 0 1- . 0 0 1- . 0 0 1+ .002- .001- .004

Fu.

000

- . 0 0 10000

TABLE 2-b

Value ofParameterwhich ischanged

X=.89ci=300c.=200c,= 70c^= 10

Cu = 1.2<;H=1 .2

6=.1O

: PRICE DcasiON

Xt-i

- . 0 1 3- . 0 0 4- . 0 0 9- . 1 3 5+ .079- .050+ .035- .135

P.-i

+ .0320

- . 0 0 1- . 0 0 5+ .030- . 0 0 1+ .001- .052

R U L E P ,

+ .001- .003

0-.039+ .031+ .024+ .001- . 0 5 1

CHANGE

Ut-i

+ .002+ .006+ .002+ .040- . 0 3 5- . 0 2 4+ .002+ .060

FROM INITIAL SET

Qt

+ .005+ .007+ .002+ .039- . 0 7 9+ .027+ .002+ .045

Qt+i

-.004+ .001+ .001+ .014- . 0 5 2- . 0 0 5- . 0 0 5

+ .011

UNDER

Qt..«

- . 0 0 800

+ .004- . 0 3 8

0- . 0 0 4- . 0 0 6

ALTERNATIVE COST

Ft_i

- .0320

+ .001+ .005- . 0 3 0+ .001- . 0 0 1+ .052

Vt

- . 0 1 700

- . 0 0 4- . 0 0 5+ .002- . 0 0 3- . 0 0 7

STRUCTURES

Ft+i

000

- . 0 0 2

+ .00500

- . 0 1 2

F,+.

+ .00100

- . 0 0 1+ .004

00

- . 0 0 7

T A B U ; 2-C: INVENTORY DECISION RULE Bt: CHANGE PROM INITIAL SET UNDER ALTERNATIVE COST STRUCTURES

Value ofParameterwhich ischanged

X=.89c, = 300C! = 200c,= 70Ci= 10

CH=1.2

6=.1O

-.001+ .006- .058+ .079- . 0 0 1- . 0 5 4- . 0 0 9+ .004

Pt-i

0+ .001- . 0 0 2+ .004

0+ .001- . 0 0 9- . 0 0 4

B

_

+—++

.0010.057.0700.065.072.002

Ut-i

+ .005*- . 0 0 2+ .098*- .115

0+ .116*+ .207*- .001

Q.

+ .004*- . 0 0 2+ .097*- .116

0+ .116*+ .205*- . 0 0 2

Qt+i

+ .001+ .007+ .006- .019

0+ .021+ .012- . 0 0 2

Qt+.

- .001+ .001- . 0 0 2- . 0 0 1+ .001+ .008- . 0 0 7- . 0 0 1

F.-i

0- . 0 0 1+ .002- .004

0- .001+ .009+ .004

Vt

0+ .001- .001+ .006

00

- . 0 0 9- . 0 0 3

F.+,

+ .0010

+ .001+ .001

0+ .002

0- . 0 0 2

000

- . 0 0 1000

- . 0 0 1

' Indicates a sign change.

540 THE AMERICAN ECONOMIC REVIEW

duction equation and the inventory equa-tion. (This was true on all trials.) Thusfor purposes of these two decisions, anorder on hand at the beginning of the per-iod is treated the same as an order anti-cipated during the period. This result is aconsequence of the particular cost struc-ture assumed since Qt and Ut-i enter thecost function symmetrically through theconstraint:

Qt - bPt -St=Ut-U,t - i

Childs (pp. 36-37) obtained the sameresult in a similar model (without a priceequation) and suggests that it demon-strates the possible over-simplification inthe U* equation, whereby the desiredlevel of unfilled orders was assumed todepend solely on current production. Hesuggests that one alternative would be tomake U* a function of Ot as well as Xtwhich would serve to break the rigid equal-ity between the two coefficients, althoughwe would not expect the signs to change.More importantly, he recommends thatin the regressions, the coefficients not beconstrained to have the same sign (i.e.,estimated as a single variable), since thiswould almost certainly involve some de-gree of specification error.

Alternatively we might think of thisresult as arising from our neglect of theproduction-to-order, production-to-stockdistinction. If, as we mentioned earlier,the desired level of unfilled orders is re-lated not to total production but only tothat portion of the total which is produc-tion-to-order, then the equality betweenthe coefficients would be broken, with Ut-ireceiving a smaller weight, especially inthe inventory equation. However, Ut-iwould stni appear in the rule for finishedgoods inventories so long as the cost ofchanges in the rate of production is a func-tion of total production and independentof the mix between order and stock.

Even assuming the plausibility of the

original cost structure, this result servesto demonstrate one of the more question-able implications of the certainty equiva-lent-quadratic criterion function approach.Ut-i is a known quantity, given to the de-cision maker before he must make his de-cisions on Xt and Ht. Ignoring the per-fect forecast, Qt is not a known quantitybut the expected value of the decisionmaker's subjective probability functionover all possible values of Q: yet for pur-poses of decisions on the rate of productionand the level of finished goods inventory,they are treated identically. Of course,this is a straightforward result of the cer-tainty equivalence theorem, and, perhapsmore suggestively, a consequence of assum-ing a quadratic criterion function. Thepoint is that this approach in some waysseems to beg the question of risk and un-certainty, and, in so doing, may lead us tooverlook what some regard as the essen-tial characteristic of unfilled orders—itsability to serve as a buffer between thepast and an uncertain future.

We also note that in the price equation,the coefficients of Ht-i and Ut-i are op-posite in sign and almost equal in absolutevalue. Hence, our model suggests that forpurposes of the price decision, it may bepossible to consider unfilled orders as"negative" inventory. However, as before,this result derives from the simple aggre-gation of order and stock production witha single price for both. In the regressionanalyses, we would expect the coefficientsof Ht-i and Ut-i to partly reflect the rela-tive weights of order and stock goods inthe aggregate price variable, and thereforenot display the same coefficient.

Finally we note that Pt-i and Ft-i havethe same absolute coefficients in all equa-tions and the relevant variable for themodel is therefore (Pt-i—^t-i). Becausethe P series is in index form, however, wedo not constrain the variables to have thesame coefficients; nor should we expect

HAY: PRODUCTION, PRICE, AND INVENTORIES 541

them to turn out equal if they are notconstrained.

III. Regression AnalysisThe Data

The data consist of monthly observa-tions for the period March 1953—August1966 on two Standard Industrial Classifi-cation (SIC) manufacturing groups—Lumber and Wood Products (SIC 24), andPaper and Allied Products (SIC 26).

The primary sources for the data aremanufacturer's shipments, inventories andunfilled orders series compiled by the In-dustry Division of the Bureau of Census,U.S. Department of Commerce,'-^" andthe Wholesale Price Index and averagehourly earnings series compiled by theBureau of Labor Statistics, U.S. Depart-ment of Labor.

As is generally known, the WholesalePrice Index is commodity-oriented, i.e.,the index is the weighted sum of the pricesfor a group of similar commodities. On theother hand, the Commerce Departmentseries are industry-oriented, i.e., the ship-ments series, for example, will include allthe shipments by a particular industrydespite the fact that the specific commodi-ties which originate in a single industrymay be quite diverse. Therefore the cover-age of a typical Commerce series is likelyto be considerably different from that ofthe most nearly related price series.

For the industries examined in thisstudy, the coverage of the two types of

' Not all the data are published by Commerce. I amgrateful to David Belsley for making available to methe series which he obtained under the stipulation thatonly the results derived from them and not the seriesthemselves be published. For a detailed description ofthe Commerce data, see Belsley's doctoral dissertation.

"• Since the theory is in terms of physical quantities,the shipments, inventories and unfilled orders variableswhich are reported by Census in value terms weredeflated by the price variable for the correspondingmonth. This procedure introduces a slight degree oferror if businesses do not value finished goods inventoryat finished goods prices.

index is reasonably close. Indeed this wasthe main reason for selecting these par-ticular industries. Hopefully, therefore,the errors introduced by the coverageproblems will be minimized, although one'sconfidence in the specific numerical resultsis somewhat weakened. It should be men-tioned that generally one can expect acloser correspondence of the two types ofdata at lower levels of aggregation. Un-fortunately, although many of the data areavailable at the 3-digit level, the series forinventories by stage of fabrication is cur-rently available only for 2-digit aggregates.

A further problem arises in attemptingto obtain estimates for the direct costs ofproduction. These are to be used in testingthe amended specification of the cost ofchanging price in which penalties are asso-ciated with price changes that are corre-lated to changes in direct costs. Althoughlabor costs can be represented by the serieson average hourly earnings, there is con-siderable difficulty in arriving at a similarindex for raw material costs. The problemarises when the output series at the 2-digitlevel includes as final product its own basicraw material input. For example, the prin-ciple raw material input for Lumber andWood Products (according to BLSweights) is lumber. Yet lumber also getsadded in as final product, and is, in fact,the largest item in that aggregate. Hencethe largest component in the price indexfor the final product is identical to thelargest component in the index of rawmaterial prices and we can expect a rathersubstantial spurious correlation betweenthe two indices. Therefore, labor costs werethe only element of direct costs used in theregression analysis. Also, due to the highdegree of collinearity in the series, onlyvalues for periods t— 1 and t were used.

Discussion of Results

Regressions were performed using simpleleast squares on both seasonally adjusted

542 THE AMERICAN ECONOMIC REVIEW

TABI.E 3—REGKESSION COErriCIENTS

(lvalues in parentlieses)

LtJMBEE

Pt

Bt

PAPER

Xt

Pt

B,

-3

5

- 0

- 1

- 0

- 0

a

.620

.584

.890

.490

.040

.250

(3

(2

(0

(0

(0

(1

.213.99)

.261

.73)

.008

.26)

.023

.75)

.019

.31)

.024

.56)

Pi-l

- . 0 0 4(0.34)

.914(39.26)

.012(1.63)

- . 0 4 3(4.34)

.970(49.20)

.004(0.80)

- . 1 0 6(1.59)

- . 3 1 8(2.65)

.914(23.98)

- . 3 0 0(3.16)

- . 5 5 4(2.90)

1.003(20.67)

Ut-i

(3

(0

(0

{6.

(1

{2-

.157.18)

.063

.72)

.017

.59)

.265,40)

.123

.48)

.05666)

Qi

.568(9.47)

.130(1.21)

- . 0 7 4(2.16)

.854(26.53)

.086(1.32)

.005(0.32)

Qt.i

.074(1.24)

.172(1.62)

.042(1.25)

- . 0 6 0(1.96)

- . 0 2 6(0.42)

.041(2.61)

Qi..

.008(0.16)

.366(4.13)

.017(0.59)

- . 0 5 6(2.03)

.049(0.89)

.006(0.39)

V

(0,

(0.

(1.

(5.'

(0.

(3."

t-i"

.006,47)

.012,47)

.01474)

,10756)

.03180)

03111)

(1

(0

Vt

.015.04)

.002.07)

- . 0 1 1(1.34)

(4

(1

(3.

.085

.10)

.046.12)

.03531)

R^

.904

.947

.902

.979

.981

.982

D.W.^

2

1

1

1.

1,

1.

.03

.42

.62

98

,80

59

* Average hourly wages in cents.•• These are presented without inference as to their significance, given the presence of a lagged dependent variable in

the equations. In addition, David Grether has pointed out that the relevant serial correlation may be other than firstorder for monthly data, e.g., £(«t, ) 0

• 150 degrees of freedom.

and nonadjusted data. The latter is re-ported below although there was very littledifference between the two. The results,which are presented in Table 3, are gen-erally consistent with the expectationsgenerated by the sample calculations.Indeed, none of the 20 coefficients whichwere predicted unambiguously by thesample calculations showed the wrong signfor both industries. Overall, out of 40 suchcoefficients, 34 were predicted correctly.Even more revealing, out of 18 coefficientswhich came out with coefficients more thantwice their standard error, only 1 was in-correct. The same regressions without anintercept term were performed on first dif-ferences. Although the ^^'s dropped con-siderably, the matching of coefficients withexpectations was not much changed.

When the results are broken down byequation they become somewhat moremeaningful. The production equation per-

formed quite well, showing the predictedsign for 14 of 18 coefficients and for 9 ofthe 10 coefficients which turned out signifi-cant. The coefficient of Zt_i is somewhatsmaller than had been predicted, especiallyfor the paper industry. This indicates thatproduction may be somewhat more flex-ible than had been assumed. If Cs weremade smaller, the effect would be to lowerthe coefficient of Zt_i and shift more ofthe weight to Ut-i and Qt. It is rather sur-prising that in the regressions on first dif-ferences, the coefficient of Jf{_i comes outsignificantly negative for both industries.This result could not be reproduced in thesimple calculations by any reasonable setof cost parameters and one is tempted toattribute it to some statistical problems(e.g., estimating first differences withoutan intercept).

The price equation is also strong with 17of 18 signs predicted correctly, and of the

HAY: PRODUCTION, PRICE, AND INVENTORIES 543

6 coefficients which turned out significant,all had the correct sign. The price equationis rather heavily dominated by the laggedprice term, and this may refiect the ten-dency of the BLS index to pick up listprices which do not necessarily refiect theterms at which transactions actually takeplace. To the extent to which this is aquantitatively significant phenomenon wewould not expect other variables to showup significantly since in the extremePt=P%-i for long periods, even thoughactual transactions prices are changing inresponse to market conditions.

The coefficient of Ht-i in the price equa-tion is considerably larger than predicted,and this result holds up in the first dif-ference regressions. No set of cost param-eters could be found which would signifi-cantly improve the predictions of thesimple calculations for this coefficientwithout throwing off some of the others;for example, increasing b makes the coeffi-cient of Ht-i larger (in absolute value) butmoves the coefficient of Xt_i in the wrongdirection. This suggests that there may besome specification error involved here.

There is little that can be said about theinventory equation since the predictionsfor most signs were ambiguous. If we judgeby the results of the initial sample calcu-lation, 14 of 18 coefficients were predictedcorrectly, including all 7 which came outsignificant. In one sense the results can beconsidered favorable since the model pre-dicted that most coefficients in the inven-tory equation would have values very closeto zero and this was borne out. The ex-tremely high coefficient of Ht-i was unex-pected, but since this result does not holdup for the first difference equations, thereis little value in trying to come up with aneconomic interpretation. Since the Durbin-Watson statistics suggest serial correlationof the residuals in the inventory equation,the value of H,-i is subject to bias, andthis might explain the result.

TABLE 4—THE IMPACT OP A ONEUNIT INCREASE IN DEMAND

Increase in productionRise in priceDrawing down

of inventoriesBuildup of

unfilled orders

Lumber

.568

.006

.074

.352

Paper

.854

.004

-.005

.147

Predicted(TABLE 1)

.492

.008

.001

.499

Impact of Changes in Demand

It is interesting to ask how the systemreacts to changes in demand; specifically,how the impact is spread over the differentvariables the firm controls—production,price, inventory, unfilled orders, and ship-ments. The answer is dependent on thecoefficient of Qt in the various equationstogether with the identity constraints in-volving those variables. The informationis summarized in Table 4 above. Eachnumber indicates what portion of a suddenand temporary one unit increase in de-mand, Q, would be borne by the variableindicated in the left-hand column .̂ ° (Theamount absorbed by a price increase isfound by multiplying the coefficient of Qtin the price equation by .05, the assumedslope of the demand curve.)

It is perhaps surprising that pricechange plays such a small role in absorbingincreases in demand although that is pre-cisely what is predicted by the model." Tosome extent this is attributable to thepreviously mentioned failure of the BLSindex to refiect all price changes, but it alsoserves to point out a possible deficiency in

" If the increase in demand is foreseen three monthsin advance, the adjustment process is only slightlydifferent.

" Table 4 understates the total contribution of pricechange even where the rise in demand is unexpectedsince price continues above its long-run equilibriumlevel for several periods. The full effect of price increasesis to absorb 11,4, and 7 percent of the demand increasein lumber, paper, and tie sample experiment, respec-tively.

544 THE AMERICAN ECONOMIC REVIEW

the model. The assumption that all costparameters remain constant throughoutthe cycle is extremely convenient mathe-matically, but almost certainly misrepre-sents the facts to some degree. For ex-ample, as capacity is approached it mustbe true that d (the cost of increasing pro-duction) rises sharply, and it is at thispoint that price increases are most likelyto play an important role. If the modelcould be amended to include factors suchas capacity utilization, the true role ofprice changes would probably show upmore emphatically.

IV. Summary and Conclusions

In this paper we have attempted to de-velop a model which explains decisions onproduction, price, and finished goods in-ventory for manufacturing firms. The key-note of the analysis was that decisions onthese variables should be treated as simul-taneous and interdependent, emphasizingthat price should not be considered asgiven, but rather as one of the variablesto be determined within the model.

To what extent can we consider themodel a "success?" It is difficult to attacha score to the overall contribution of theanalysis but some observations are pos-sible. First, we have demonstrated that itis technically possible to treat the decisionvariables as being determined simul-taneously, and that given certain assump-tions with respect to the demand curve,there is no reason why the price decisioncannot be included. Second, the model hasthe desirable property that it is capableof generating empirically testable (andtherefore refutable) hypotheses under mostconditions. This may explain why as yetthere has been relatively little enthusiasmfor many of the nonmaximization theorieswhich, although perhaps appealing to ourintuitive feelings about the workings of thebusiness world, have not, for the most part,yielded substantive implications with re-

gard to observable phenomena. Finally,the model seems to hold up reasonablywell when applied to selected 2-digit indus-try groups, despite obvious problems withthe data. In particular we noted the im-perfect matching of the BLS price seriesto the 5/C-based production series, andthe possibility that even the "correct"price list might not always refiect the ac-tual terms of a transaction.

The final point to be made regards thedirection of future research in this area.Certainly there is still much work to bedone within the framework of the currentmodel. Some different specifications mightbe tried, for example making H* dependon new orders as well as shipments andsimilarly for U*. Alternative lag structurescould be tried for the direct cost variablesand the treatment of expectations aboutdemand might be improved. We mighteven want to get behind the productionvariable by actually including a productionfunction in the model. This might enableus to derive decision rules for the size ofthe work force, amount of raw materials,and inventories at various stages of fabri-cation. Finally, we might expand themodel to include some of the less easilyquantified variables such as advertisingand other aspects of nonprice competition.

In many ways, however, the most criti-cal need is for better data. The individualproblems have been discussed above andit is not necessary to repeat them here.However the point must be stressed that itis only with data which are accurate,consistent, and appropriate to the levelof aggregation of the model that anytheory in this area can be given a fair andrigorous test.

REFERENCESD. A. Belsley, "Industry Production Behavior:

An Economic Analysis," unpublished doc-toral dissertation, M.I.T. 196S.

G. L. Childs, Unfilled Orders and Inventories:a Structural Analysis, Amsterdam 1967.

HAY: PRODUCTION, PRICE, AND INVENTORIES 545

P. Darling, and M. Lovell, "Factors Infiuenc-ing Investment in Inventories," in Dusen-berry et al., eds.. The Brookings QuarterlyEconometric Model of the U.S., Chicago1965.

O. Eckstein, and G. Fromm, "The Price Equa-tion," Amer. Econ. Rev., Dec. 1968, 43,1159-83.

W. Fellner, Competition Among the Few, NewYork 1960.

C. C. Holt, F. Modigliani, J. F. Muth, andH. A. Simon, Planning Production, Inven-tories, and Work Force, Englewood Cliffs1960.

E. S. Mills, Price, Output and Inventory Policy,New York 1962.

B. R. Moss, "Industry and Sector Price In-dices," Mon. Lab. Rev., U.S. Department ofLabor, Aug. 1965.

H. A. Simon, "Dynamic Programming underUncertainty with a Quadratic CriterionFunction," Econometrica, Jan. 1965, 24,74-81.

H. Theil, Linear Aggregation of Economic Re-lations, Amsterdam 1954.

, "A Note on Certainty Equivalence inDynamic Planning," Econometrica, Apr.1957, 25, 346-49.