Journal of Productivity Analysis, 11, 55–66 (1999)c© 1999 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.

Scale Economy Measures andSubequilibrium Impacts

CATHERINE J. MORRISON PAULUniversity of California at Davis, Department of Agricultural and Resource Economics

Abstract

Morrison (1985), Morrison and Siegel (1997) and Morrison and Schwartz (1994) have sug-gested using an expression for “total” scale or cost economies to disentangle determinantsof cost efficiency, including short run subequilibrium effects. Fousekis (1998) has notedthat the derivation of such an expression is based on imputation of the long run, whichimplicitly suggests evaluation at steady state values. Measurement of elasticities imputingvalues not observed in the data, however, invariably requires some type of approximation.The Fousekis approach represents one view of the relevant approximation, which is notconceptually appropriate for most applications, and disallows evaluation of the implied ad-justment process to long run values. This article highlights the underlying assumptions thatraise questions about this approach, and overviews alternative approaches to and rationalesfor computing these types of elasticity estimates.

Keywords: costs, scale economies, utilization, short and long run, subequilibrium

I. Introduction

The cost-output relationship is of crucial importance in cost and production analysis. Thisrelationship is typically assumed to represent the cost-minimizing method of producing anyparticular output level at existing technology levels and input prices. In response to changesin economic conditions, the firm must re-optimize to reach a new cost minimization point.As recognized by Marshall more than a century ago, if the adjustment process takes sometime the resulting short run cost-output relationship will differ from that attainable in thelong run.

Measurement of short and long run scale economies requires representing these cost-output relationships effectively. The implied adjustment process initially involves a shortrun response, which depends on the shape of the underlying short-run cost curve. Ultimately,adjustment along the long run cost curve will occur, as the firm adapts its quasi-fixed inputlevels. Other external factors may also impact the observed cost-output relationship byshifting instead of creating a movement along the existing cost curves.

Empirically untangling these responses based on observed short run or subequilibriumdata involves identifying the short run response, and imputing the implied long run behav-ior. This imputation requires some type of approximation, since the observed values by

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definition do not reflect long run equilibrium. Determining the appropriate approximationfor a particular application requires careful conceptualization of how adjustment processesoccur, and consideration of where evaluation of long run measures should be carried outwhen based on discrete data.

One approach to this, developed and used by Morrison (1985), Morrison and Schwartz(1994) and Morrison and Siegel (1997), derives the long run cost-output relationship byviewing potential long run adjustment from the observed short run point. The resultingexpression appends an adjustment factor representing the extent of subequilibrium to a shortrun cost-output elasticity reflecting observed behavior. The movement toward a steady stateis thus decomposed into short and long run components.

Fousekis (1998) has instead argued that the adjustment factor should be evaluated at animputed long run based on imposing the envelope condition, so it vanishes by construction.Imposing the envelope condition at current capital and output levels, however, implicitlyrequires revaluing the capital stock so that these levels are consistent with long run equi-librium. This approximation is not conceptually appealing for empirical application, sincelong run behavior implies adjustment of current stock levels given observed prices ratherthan revaluation of the price of the capital stock. In addition, it does not allow considerationof determinants underlying the adjustment process, since it evades the critical question ofhow to measure and interpret thedifferencebetween the short and long run cost-outputrelationships or scale economies.

The purpose of this article is to overview the conceptual basis, underlying assumptions,and empirical evidence associated with different approximations of long run behavior fromshort run data. The combined elasticity approach used by Morrison and others offers atheoretically and empirically useful representation of the long run from the perspectiveof the short run. This approach also allows evaluation of the determinants and extent ofsubequilibrium. By contrast, the Fousekis approach, although theoretically consistent, lacksa conceptual rationale and does not facilitate interpretation.

II. The Simple Economics of Subequilibrium Adjustments

Standard cost curves from intermediate microeconomics represent minimum productioncosts for any output level. Input fixities cause short run costs to exceed costs in the longrun, when all inputs can be varied, except at the point of tangency of the curves. Thesenotions are typically illustrated in terms of the average and marginal cost curves, as inFigure 1.

For one output,Y, and one capital (quasi-fixed) input,K , the short run cost curve isdrawn in terms of the existing capital stock levelK0. The optimal or capacity output levelfor K0 (the steady state output associated with available capacity determined byK0) isY0 = Y∗(K0). If the firm is in short run subequilibrium, however, output demand may besuch that the firm is producing outputY1 with capitalK0 (point B), and thus is incurringhigher costs than would be possible in a steady state. The firmís response would then be toadjustK in the long run to its “desired” capital levelK1, or K ∗(Y1) (pointC).1

The long run cost curve, reflecting cost-minimizing input demand behavior for the existingtechnology and factor prices, is represented by the cost functionC(Y,p) (wherep is a vector

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Figure 1. Short and long run cost curves.

of all input prices including the market price ofK , pK ). The short run curves are insteadrepresented byG(Y, K ,p)+ pK K , wherep includes only variable input prices.

Technical change (typically included as a time counter, “t”) or other environmental orexternal factors may also appear as arguments of the total and variable cost functionsC(•)andG(•). These factors shift rather than cause movement along the curve, since they alterthe production functionY(K , v) (wherev is a vector of input quantities). Such externalfactors will be ignored below, to focus on the issue of short-to-long run adjustment, but thisdoes not affect the substance of the arguments.

G(Y, K ,p) + pK K exceedsC(Y,p) unlessK = K ∗. The short run average cost curve(SRAC) is thus more steeply sloped than the long run curve (LRAC) except at the optimaltangency point, and short run marginal cost (SRMC) is lower (forK > K ∗) or higher(for K < K ∗) than long run marginal cost (LRMC). A clear relationship therefore existsbetween the cost-output relationship—and thus “scale” economies embodied in the slopesof the average cost curves—in short run subequilibrium and long run equilibrium. If wewish not only to measure the short run cost-output relationship from observed data (at pointB), but also to impute what it might be when long run adjustment takes place (pointC), thelong run relationship must be approximated.

This may be accomplished by appending a term representing the subequilibrium impactto the short run measure. This allows the determinants of the distance, or wedge, betweenthese points to be characterized. In particular, Morrison (1985) suggested that the long run

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cost-output or scale relationship may analytically be represented as

εLCY = εS

CY + εC KεKY, (1)

whereC = CS = G(Y, K ,p)+ pK K , εSCY = ∂ ln C/∂ ln Y, εC K = ∂ ln C/∂ ln K , εKY =

∂ ln K ∗/∂ ln Y, S indicates “short run”, andL represents “long run”.2

This equation is based on the idea that the desired long run level of the capital stockK ∗, asviewed from the observed short run point, may be imputed from information about the extentof subequilibrium built into the model. That is, a measure of the short run shadow value ofK , or the variable-input-cost-saving from an additional unit ofK , may be constructed asZK = −∂G/∂K . Long run equilibrium is thus obtained whenK adjusts to the point wherethis marginal benefit is equal to the marginal cost ofK ; ZK = pK .

Imposing this equality and solving for the impliedK ∗ (which is possible with a flexiblefunctional form sinceZK is then a function of all arguments of theG(•) function) results inthe expressionK ∗(p,Y, pK ). In the Figure this reproduces the capital level consistent withthe tangency of SRAC and LRAC at the given output level,Y1 (point C). On an isoquantdiagram, this would be theK level corresponding to the point on theY1 isoquant where theisocost curve defined in terms of the market pricepK is tangent.

The εC K measure in (1) represents the extent or costs of the short run subequilibriummotivating this adjustment process, through the deviation betweenpK andZK , sinceεC K =∂ ln C/∂ ln K = ∂C/∂K (K/C) = (pK − ZK )(K/C) (from the definitions ofC = CS andZK above). TheεKY elasticity, capturing the capital investment response to output changes,may be computed directly from theK ∗(p,Y, pK ) expression (if theG(•) function is suchthat it can be solved for analytically).

Evaluating these measures at the observed short run values ofK (K0),Y(Y1) and pK

provides insights about the adjustment process to the long run from the short run observedpoint. Alternatively, however, the long run might directly be imputed and the elasticitiesevaluated at this point. Such an approximation requires measuring the implied steady statevalue ofK , Y, or pK , in order to evaluate the elasticities at this value.

That is, subequilibrium implies that the observedY, pK , andK values are not consistentwith a steady state. A steady state may therefore be imposed by evaluating costs, and theassociated cost elasticities, atY∗(K0) instead ofY1, ZK instead ofpK , or K ∗(Y1) insteadof K0.

The first of these possibilities provides insights about capacity utilization, since it involvesidentifying the optimal or potential output level (in the sense of a steady state) consistent withexisting capacity. However, it is not conceptually appropriate for representing adjustment,since one would not think firms adapt output production to fully utilize capital, but insteadadjust capital inputs according to output demand.

The second possibility, evaluation atZK instead ofpK , imposes the envelope condition atobservedK ,Y levels. It thus provides information about what capital price would supportthe existingK -Y combination as a steady state relationship. This does not, however, providea valid conceptual basis for representation of long run adjustment. The firm does not revalueits capital stock, but instead invests according to the market price of capital it perceives.The former notion is like repositioning the isocost curve in an isoquant diagram to force a

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tangency for a given output and capital level. It reproduces a long-run equilibrium, but notthe equilibrium the firm will adjust toward.

The final approximation to the long run involvesK adjustment. IfK ∗(Y1) is imputedby settingZK (Y1) = pK and solving for the equilibrium value ofK , this represents theadjustment process we conceptualize in economic theory. The firm assesses output demandand the market price of capital, and makes long term investment plans accordingly. Thislong run is viewed, however, from the perspective of the existing short run point, and thusis based on observed current economic conditions.

Since the first of these alternative long run approximations—measurement of and evalua-tion atY∗ instead ofY—is typically only used in the context of capacity utilization measures,we will pursue further only the second and third approaches. For these cases costs maybe expressed asC∗Z = G(•) + ZK K (shadow costs), andC∗K = G(•, K ∗) + pK K ∗ =C(p, pK , K ∗,Y) = C(p, pK ,Y), respectively. Computed elasticities based on these ap-proximations to long run costs will by definition collapse to their steady state values. Thus,εC K = 0, andεCY may by construction be interpreted as a long run measure.

However, as noted above, the conceptual basis for the shadow cost approach is question-able for purposes of empirical implementation and interpretation. In addition, representingthe determinants and extent of subequilibrium, and thus evaluating the implied adjustmentprocess from observed short run data, is not facilitated by these approximations. By con-trast, the combined elasticity approach resulting in equation (1) allows interpretation ofthe wedge betweenεS

CY andεLCY since it focuses on adjustment from the viewpoint of the

observed short run subequilibrium point.All these measures are approximations from different perspectives. It is inevitable that

some approximation is necessary when empirically imputing expected long run behaviorfrom observed short run responses. Also, applying a theoretical model to discrete data doesnot involve continuous changes that are readily analyzed by procedures such as imposingan envelope theorem, as touted by Fousekis (1998). If one is to choose among the approx-imations, it is thus important to consider the conceptual rationales for, and empirical andinterpretive power of, the different methods.

In particular, it is worth emphasizing once again that appropriate imputation of the longrun conceptually involves adjustment of the capital input to its desired level, rather thanrevaluing the capital stock to reproduce some theoretical steady state. This could involveimputing the long runK ∗ and evaluating the elasticities at that point, or viewing long runtowardK ∗ from the vantage point of short run observed behavior.

These two approximations would be equivalent if the data were in continuous rather thandiscrete form. This can be formally supported by deriving expression (1), evaluated at theexisting K0,Y1 levels, as a theoretical approximation to theεL

CY measure resulting fromtheC∗K representation of costs, as in the following subsection. In addition, the empiricalapproximation of the combined elasticity in (1) to that measured directly viaC∗K is veryclose, as is shown in the next subsection. However, the measure based onC∗Z differs, bothconceptually and empirically, from either of these measures.

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III. A Theoretical Approximation

The decomposition of theεLCY expression in (1) relies on specifying the desired level of

capital as a function of all arguments ofG(•), includingY. Thus long run total costs canbe written asC∗K (Y,p, pK , K ∗(Y; (•)), and∂C∗K /∂Y = ∂C/∂Y + ∂C/∂K (∂K/∂Y) bythe chain rule. This suggests that theεL

CY equation should be evaluated at thisK ∗ levelif one wishes to impose the imputed long run. However, as noted in the previous section,this approach bypasses a primary issue of interest in this type of analysis—what drives thewedge between short and long run costs, which motivates capital adjustment from short runobserved values.

If εCY is evaluated atK ∗, which must in turn be imputed, it simply reduces to the long runmeasure with the adjustment termεC KεKY equal to zero. This leaves open the question ofhow one might analytically represent the difference betweenεCY evaluated atK0 (point B)and atK ∗(Y1) = K1 (point C). It is desirable to approximate this deviation from the vantagepoint of the short run observed levels.

In certain cases—in particular if the functions are linear so theZK (and thusεC K ) measuredoes not depend onK—the approximation is exact. If this is not the case, which is generallytrue for flexible functional forms which capture interactions among inputs, (1) can bemotivated as a (continuous) first order Taylor series approximation to the long run measurefrom C∗K , evaluated at the short run levels ofK andY. This involves expanding around thelong run cost levelC∗K (evaluated atK ∗(Y1)):

C|K ∗(Y1),Y1 ≈ C|K ∗(Y0),Y1+ ∂C/∂K |K ∗(Y0),Y1∂K ∗/∂Y (Y0) · [Y1− Y0], so, (2a)

C|K ∗(Y1),Y1− C|K ∗(Y0),Y1

≈ ∂C/∂K |K ∗(Y0),Y1∂K ∗/∂Y (Y0) · [Y1− Y0], or, (2b)

[C|K ∗(Y1),Y1− C|K ∗(Y0),Y1]/[Y1− Y0]

≈ ∂C/∂K |K ∗(Y0),Y1∂K ∗/∂Y(Y0). (2c)

This approximates the gap between the long run and short run values of∂C/∂Y:

dC/dYK ≈ ∂C/∂K |K0,Y1 ∂K ∗/∂Y, (3)

wheredC/dYK , a nonzero approximation to∂C/∂K |K∗(Y1),Y1, stems from capital ad-justment from the existing capital levelK0 = K ∗(Y0). Further, this can be rewritten inproportional or elasticity terms as:

d ln C/d ln YK ≈ ∂ ln C/∂ ln K |K0,Y1 ∂ ln K ∗/∂ ln Y = εC KεKY (4)

as in (1) (forεLCY − εS

CY).This expression approximates the actual adjustment, evaluated at observed short run

subequilibrium values and presented in continuous terms. However, as noted above, itprovides further information that is trivialized by the imposition of the long run by theenvelope condition. It comprises an analytical expression for the deviation between thelong and short run values that has a useful interpretation in terms of the costs of capitalfixity (εC K) and desired capital adjustment based on current output demand(εKY).

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TheεLCY expression in (1) thus identifies the factors contributing to the deviation between

short and long run costs. It is a continuous approximation from the perspective of theobserved short run subequilibrium point. However, the necessity for approximations arestandard when empirically computing elasticity measures, since with discrete data they maybe evaluated beforeor after the underlying implied change occurs.

IV. An Empirical Approximation

The previous sub-sections have addressed the conceptual and theoretical basis for imputinglong run behavior from short run observed data by different approximations. However,since the fundamental issue of measuring elasticities is one of empirical application, it isparticularly important to consider the associated empirical implications.

Elasticities of this sort have been used in various contexts in the literature. One recentapplication of these procedures is to measurement of cost economies in the meat packingindustry (Morrison, 1997). This application recognizes both quasi-fixed capital and thepotential deviation of livestock shadow values from their market prices due to marketpower, accommodated by sequential optimization.

In this context, short run costs may be written asCS = G(p,Y,M, K )+pM(M)M+pK K ,“intermediate run” costs areCI ∗

M = G(p,Y,M∗, K ) + pM(M)M∗ + pK K , and long runcosts areCL∗

K = G(p,Y,M∗, K ∗)+ pM(M)M∗ + pK K ∗ (whereM is materials, which isprimarily livestock).3 TheεCY elasticities associated with these three different “runs” maybe computed either by substituting theM∗ andK ∗ functions and evaluating the resultingexpressions at theseM, K levels (theC∗K method), or using combined elasticity formulas(as in (1)), including terms for bothM andK adjustment. Another possible alternative isthe revaluation approach, that imposes the envelope condition at givenM andK levels byvaluing these inputs at their shadow values (theC∗Z method).

Short- intermediate- and long-runεSCY, ε

ICY and εL

CY elasticities based on these threemethods are presented in Table 1. Measures computed using the revaluation procedureto impute intermediate or long run costs are identified by aC∗z subscript, those basedon M∗, K ∗ substitution are denoted byC∗k subscript, and those constructed as combinedelasticities are reflected by aC∗c subscript.

Although the elasticities do not vary dramatically, those based on the substitution andcombined elasticity methods (the third and fourth column, and the last two columns) cor-respond the most closely. Elasticities based on the revaluation or shadow valuation method(second and fifth columns) exhibit somewhat different patterns.

All approximations are similar in the sense that they indicate the presence of scaleeconomies in most years throughout the sample. The statistical significance differs, how-ever, when standard errors are computed for the elasticity measures (theC∗z elasticities donot vary as much and are not as often significantly different from one). Also, the evidenceof decreasing scale economies in the intermediate run toward the end of the sample is notas definitive using the revaluation method.

Some variation among these different approximations would be expected, since the con-ceptual basis for and thus interpretation of the measures differ. However, the combined

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Table 1.Short, intermediate and long run scale elasticity measures.

Year εSCY ε I

C∗zY ε IC∗kY ε I

C∗cY εLC∗zY εL

C∗kY εLC∗cY

1965 0.933 0.972 0.996 0.997 0.957 0.974 0.9771966 0.888 0.978 1.039 1.040 0.960 1.020 1.0141967 0.896 0.955 0.995 0.995 0.943 0.979 0.9791968 0.859 0.935 0.990 0.991 0.932 0.988 0.9871969 0.875 0.963 1.025 1.026 0.951 1.014 1.0101970 0.942 0.969 0.986 0.988 0.960 0.971 0.9761971 0.952 0.951 0.948 0.953 0.956 0.950 0.9601972 0.835 0.954 1.036 1.041 0.947 1.034 1.0311973 0.691 0.913 1.116 1.194 0.951 1.166 1.2261974 0.841 0.966 1.046 1.052 0.963 1.051 1.0491975 0.918 0.988 1.029 1.029 0.987 1.030 1.0281976 0.942 0.961 0.970 0.971 0.948 0.951 0.9551977 1.089 0.989 0.940 0.955 0.959 0.883 0.9111978 0.943 0.987 1.008 1.008 0.960 0.976 0.9721979 0.826 0.961 1.035 1.045 0.948 1.034 1.0301980 0.908 0.971 1.002 1.003 0.950 0.981 0.9751981 0.972 0.980 0.984 0.984 0.950 0.944 0.9431982 0.912 0.960 0.983 0.983 0.944 0.966 0.9631983 0.990 0.974 0.967 0.969 0.946 0.928 0.9321984 0.955 0.975 0.984 0.984 0.952 0.957 0.9561985 0.991 0.964 0.953 0.955 0.949 0.929 0.9361986 0.955 0.962 0.965 0.966 0.947 0.946 0.9481987 0.903 0.978 1.011 1.013 0.954 0.991 0.9841988 0.905 0.982 1.015 1.016 0.963 1.000 0.9941989 0.983 1.013 1.025 1.025 0.986 0.996 0.9931990 0.877 0.984 1.030 1.035 0.982 1.036 1.0341991 0.982 1.006 1.016 1.016 0.980 0.987 0.985

elasticity approach seems to fare better than the revaluation method for empirical represen-tation of the long run, as well as the conceptual rationale.

V. Utilization and Subequilibrium Measurement

A final issue raised by Fousekis (1998) about theεC KεKY adjustment term is whetherthe sign of this expression unambiguously captures the difference between the short andlong runεCY values. In particular, ifεC K = (pK − ZK )K/C is interpreted as a capacityutilization measure, Fousekis argues that whetherK is over- or under-utilized(εC K < 0 orεC K > 0) does not unambiguously determine whetherεL

CY exceeds or falls short ofεSCY.

This argument is misleading, however, at least for the restricted cost function case.We can motivate this using Figure 2. Figure 2a represents the situation with constant

returns to scale (CRTS). Although in this caseεLCY = 1, in subequilibrium there is a clear

relationship between the slopes of the short and long run average cost curves, and thusεSCY

andεLCY. The slope of the SRAC curve is negative to the left of the tangency (short run

economies) and positive to the right (short run diseconomies).Following the arguments above regarding equation (1), this difference depends onεC K .

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Figure 2a.Cost curves and utilization determination: constant returns to scale (CRTS).

To the left of the tangency capital is underutilized soZK < pK , εC K > 0, andεSCY =

εLCY − εC KεKY < εL

CY. The reverse occurs on the right hand side.With nonconstant long run returns to scale (in Figure 2b), the relative relationship between

the slopes is maintained; to the right of point A there are relatively diminishing returns inthe short run, and the reverse is true on the left. Thus there is a definitive linkage betweentheεC K measure (negative or positive as output is to the right or left of the tangency point),and the relationship betweenεL

CY andεSCY.

The discrepancy between this simple graphical derivation of the problem and that pre-sented in Fousekis (1998) primarily arises because the analysis must be based on total ratherthan variable costs. AlthoughG(•) is strictly decreasing inK , C(•) is not. If K > K ∗

the marginal benefit of an increase inK (ZK ) falls short of the cost of the investment(pK ),increasing net costs as represented byεC K , even thoughZK is always positive. This breaksthe link motivated in terms of the monotonicity condition in the Fousekis treatment.

This can be elaborated following the essence of the Fousekis argument, but in a moretransparent form. Consider Figure 2b in the case of under-utilization(Y < Y0), whereSRMC= E, LRMC= D, SRAC= B andLRAC= C. Thus, althoughE < D, consistentwith Fousekis,B > C in terms of total costs. Costs at the given output level are such thatSRAC> LRAC.4 Ambiguity about the relative sizes of theεL

CY = H/G andεSCY = I /F

measures is thus avoided;5 the gap betweenE andB must exceed that betweenD andC.However, this does not alone seem to reconcile the problem. In the case of over-utilization

(Y to the right ofY0), SRMC= I , LRMC= H, SRAC= F andLRAC= G. Thus, since

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Figure 2b.Cost curves and utilization determination: nonconstant returns to scale (NCRTS).

I > H and F > G, without additional information there could be ambiguity about therelative sizes ofεL

CY = H/G andεSCY = I /F . Reconciling and interpreting this is facilitated

by specifying the problem in the context of required relationships betweenMC andAC.Assessing this issue in terms ofMC andAC when both are higher or both lower than the

comparison point buries the fundamental cost relationshipMC = AC+ Y∂AC/∂Y.6 Thisexpression can be used to shed further light on the graphical treatment.

First, note that the discussion above has been specified in terms of relative slopes of theACcurves, and theMC to AC relationship. Thus, in theCRTScase of Figure 2a,∂AC/∂Y = 0for the long run cost curve (MC = AC) but ∂AC/∂Y < or > 0 to the left and right handside of the tangency. These cases correspond toMC/AC= [AC+Y∂AC/∂Y]/AC< (>)1,respectively, and thusεS

CY < (>)εLCY.

Similarly, this expression can be used to expand on the relationship betweenεLCY =

H/G and εSCY = I /F above. From the definitionsεS

CY = SRMC/SRAC= (SRAC+Y∂SRAC/∂Y)/SRACandεL

CY = LRMC/LRAC = (LRAC+ Y∂LRAC/∂Y)/LRAC, it isclear that the deviations of these measures from one (and thus the relative sizes of the shortand long run elasticities) are determined by∂SR AC/∂Y and∂L R AC/∂Y (the slopes ofthe short and long run average cost curves). These slopes are in turn determined by theextent short run costs exceed long run costs, and thus by the deviation ofZK andpK or thesign of theεC K elasticity, in terms of total costs.7

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VI. Concluding Remarks

Theoretical and particularly empirical analyses of scale economies and fixities or subequi-librium must be approached thoughtfully. A number of potential confusions for imple-mentation and interpretation of these models arise when applying the theoretical models todiscrete short run data.

In particular, measurement of long run elasticities requires some type of approximationsince the long run must be imputed from short run observed data. The usual conceptual basisfor representing long run behavior imputes capital adjustment from the short run towardlong run equilibrium values. Morrison and others have accordingly modeled long run scaleeconomies in terms of a combined elasticity, that augments the short run elasticity by anadjustment factor identifying the extent and determinants of the costs of subequilibriummotivating long run adjustment.

This approach facilitates interpreting the causes of the wedge between measured shortrun and long run elasticity values, and theoretically and empirically approximating theassociated adjustment of quasi-fixed inputs as perceived from the short run. It thus hasconceptual and interpretational advantages over the revaluation of quasi-fixed input pricesto reproduce the long run, as implied by imposing the envelope condition at observed outputand quasi-fixed input levels.

Notes

1. Similarly, note thatK0 can be expressed asK ∗(Y0).

2. If other quasi-fixed private factors, adjustment costs, or external factors are also included in the analysis thiseasily generalizes, as in Morrison and Siegel (1997).

3. The function used in this paper also includes adjustment costs forK and technological change (shift) factorsas arguments of the function, but this does not affect the applicability of the analysis for this problem. SeeMorrison (1997) for further details about the model and measurement procedures.

4. In the Fousekis analysis, due to the monotonicity condition∂G/∂K < 0 (although his more rigorous treatmentis in terms of a dynamic value function), he implies thatC > B.

5. Note thatεCY = ∂ ln C/∂ ln Y = ∂C/∂Y(Y/C) = MC · Y/AC · Y = MC/AC.

6. This equation is often derived in intermediate microeconomics textbooks by taking the derivative with respectto Y of theAC= C/Y equation, and reorganizing.

7. This basic derivation does not incorporate some aspects of the flow dynamic analysis overviewed by Fousekis.Elaboration of the problem in the context of a flow analysis, when adjustment costs may appear even in the longrun in the form of amortized costs, affect the graphical illustration. However, the fundamental relationshipsare maintained.

References

1. Fousekis, Panos. (1998). “Temporary Equilibrium, Full Equilibrium and Elasticity of Cost.”Journal ofProductivity Analysis, this volume.

2. Morrison, Catherine J. (1985). “Primal and Dual Measures of Economic Capacity Utilization: An Appli-cation to Productivity Measurement in the U.S. Automobile Industry.”Journal of Business and EconomicStatistics4, 312–324.

3. Morrison, Catherine J. (1997). “Cost Economies and Market Power in the U.S. Meatpacking Industry.”Manuscript, April.

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4. Morrison, Catherine J., and Amy Ellen Schwartz. (1994). “Distinguishing External from Internal ScaleEffects: The Case of Public Infrastructure.”Journal of Productivity Analysis5(3), 249–270.

5. Morrison, Catherine J., and Donald Siegel. (1997). “External Capital Factors and Increasing Returns in U.S.Manufacturing.”Review of Economics and Statistics79, 647–655.