PRODUCTIVITY GROWTH, SCALE ECONOMIES, AND CAPACITY UTILIZATION IN U.S. RAILROADS, Douglas W. Caves Laurits R. Christensen Joseph A.

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1 H PRODUCTIVITY GROWTH, SCALE ECONOMIES, AND CAPACITY UTILIZATION IN U.S. RAILROADS, o Douglas W. Caves Laurits R. Christensen Joseph A. Swanson < o c d S RESEARCH NORTHWESTERN UNIVERSITY EVANSTON ILLINOIS

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3 PRODUCTIVITY GROWTH, SCALE ECONOMIES, AND CAPACITY UTILIZATION IN U.S. RAILROADS, Douglas W. Caves Laurits R. Christensen Joseph A. Swanson January 1980

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7 PRODUCTIVITY GROWTH, SCALE ECONOMIES, AND CAPACITY UTILIZATION IN U.S. RAILROADS, * * f Douglas W. Caves, Laurits R. Christensen, Joseph A. Swanson Numerous studies, including Solow (1957) and Jorgenson and Griliches (1967), have emphasized the importance of distinguishing between movements along a production function and shifts In a production function. Such shifts have been given various labels including productivity growth, technological change, and technical progress. In this paper we refer to the shift of a production function as productivity growth. Estimates of productivity growth, and hence attempts to distinguish productivity growth from movements along the production function, depend heavily on the assumptions made regarding the structure of production and the behavior of the firm. Studies of productivity growth typically assume that observed data reflect a constant returns to scale structure of production and long-run equilibrium for the firm. If these assumptions are violated, then estimates of productivity growth include the effects of scale economies and movements toward or away from long-run equilibrium, in addition to shifts in the structure of production. The first objective of this paper is to propose methodology for obtaining estimates of productivity growth which do not reflect * Economics Department, Unxversity of Wisconsin-Madison. Graduate School of Management, Northwestern University. The authors are grateful to the National Science Foundation for research support, to Philip Schoech, Michael Tretheway and Mario Miranda for proficient research assistance, and to Gary Chamberlain for helpful discussions. 1See Solow (1957, p. 312) and Jorgenson and Griliches (1967, p. 254)

8 changes in scale and movements toward or away from long-run equilibrium. Our first task in developing this methodology is to provide an unambiguous definition of productivity growth. In particular we seek a definition which permits us to study productivity growth in the case of a general structure of production with multiple inputs and multiple outputs. The second task in our methodological development is to demonstrate how the measurement of productivity growth can be approached from either the primal or the dual characterization of the production structure, even when some inputs are not employed at long-run equilibrium levels. The second objective of this paper is to apply our methodology to obtain.estimates of productivity growth for Class I U.S. railroads in the period This application is attractive because the assumptions of constant returns to scale and long-run equilibrium have both been questioned for U.S. railroads in the postwar period. Numerous Class I railroads were so small that they were unlikely to have exhausted available scale economies by In the following years some of these railroads grew substantially, both via increases in freight traffic and via mergers and consolidations, and thus had the possibility of exploiting economies of larger firm size. It is widely believed that U.S. railroads have operated with substantial excess capacity. The recent growth in freight traffic has been accompanied by abandonment of track, which suggests that the railroads have been moving in the direction of long-run equilibrium

9 Our estimates of productivity growth are derived from estimates of the railroad cost structure. There is a well established tradition of using cost function analyses to model the railroad industry. This practice is predicated on the fact that railroads are common carriers: They are required to accept all traffic at rates approved by the Interstate Commerce Commission, and hence their output levels can be taken as exogeneous. In our application we use the multiproduct variable.eost function to estimate the structure of railroad production. Use of the variable cost function permits us to specify that railroads may be in disequilibrium with respect to their stocks of way and structures. We use the generalized trans- 2 log functional form to represent the variable cost function. form imposes no restrictions on the degree of returns to scale in the production structure. We estimate the parameters of this cost function using cross-section data from 1955, 1963, and The implied estimates of productivity growth for the typical Class I This railroad are 3.5% per year, , and.6% per year, For the full period we estimate an average annual rate of productivity growth of 1.8% per year. For comparison we have also estimated productivity growth under the assumption of long-run equilibrium. Under the specification of long-run equilibrium the estimate of productivity growth for the full period is.9% per year. The and estimates are 1.0% and.5% per year. 2 See Caves, Christensen, and Tretheway (1980)

10 II. Methodology The transformation function for a general structure of production can be written: (1) F( n Y,,..., In Y, In Xn,..., In X, t) = 1; J. m 1 n where the Y, are outputs, the X. are inputs, and t is time. The total differential of (1) is: (2) EFV. d In Y. + EFY, d In X. + F.dt = 0, where the FYi are partial derivatives of F with respect to the tn Y.» etc. In the single output case productivity growth is typically defined as the rate at which output can grow over time with inputs held constant: 8 In Y/3t. With multiple outputs the natural generalization is to define productivity growth as the common rate, at which all outputs can grow over time with inputs held fixed: d In Y±/dt = d In Y /dt; d In X = 0. This definition of productivity growth can be expressed in terms of the partial derivatives of the transformation function: <3> ~dt- d In Y. d In X. = 0 x Ft EFYi In the raultiproduct case it is just as natural to define productivity growth as the common rate at which all inputs can be decreased over time with outputs held fixed: -d tn X./dt = x

11 (5) r = d ÙI X. dt = 0 - d Zn X./dt: d Zn Y - 0. This results in an alternative formula for 3 productivity growth: (A) d Zn \ dt d Zn Y± = 0 TV "x± It is straightforward to show that the two expressions for productivity growth, (3) and (4), are related by the degree of returns to scale in the transformation function. We define returns to scale as the proportional increase in all outputs resulting 3 from a proportional increase in all inputs with time fixed: Combining (5) and (2) we find (6) r = d Zn Y. d Zn X dt = 0!FYi ' which implies the following relationship between (3) and (4) (7) d Zn X. x dt d Zn Y d In 1 r dt Y. i d Zn X See Baumöl (1975) and Panzar and Willig (1977) for discussion of returns to scale in the multiproduct case.

12 Equation (7) demonstrates that our alternative definitions of productivity growth are equivalent if and only if r = 1, in which case the transformation function exhibits constant returns to scale in the region where productivity growth is being evaluated. If the firm minimizes cost with respect to all inputs, and there is a convex input structure, then there exists a total cost function, dual to (1), which can be written: (8) In CT = G( n Y,..., In Y, UP.,..., In P, t), 4 where the P. are the prices of the X. and CT = Z P.X.. From knowi x 11 ledge of the cost function (8) we can infer the structure of the transformation function (1). In particular, comparative statics analysis a la Samuelson (1947) implies the following relationships between the derivatives of (1) and (8): (9) 9 In CT/9 ZnY=- FV./ZFV., i Yi Xi (10) 3 In CT/3t = - F /EFV.. t Xi Combining (9) with (6) provides the formula for computing returns to scale from the total cost function: (11) r = (E(3 In CT/9 In Y.))"1. 4 See McFadden (1978)

13 Combining (10) with (3) provides the formula for computing productivity growth, conditional on constant output levels, from the total cost function: (12) d In X. l dt d In Y. = 0 x 3 In CT 8t Finally, combining (7), (11) and (12) we find that productivity growth conditional on constant input levels, definition (4), is 5 given by: d In Y. M TS 1 (13) dt d In X. = 0 l 9 In CT v 9 In CT 3t // " 9 In y± If the firm does not minimize cost with respect to all inputs, the cost function (8) does not exist. However, if the firm minimizes the cost of a subset of inputs (variable factors) conditional on the levels of the remaining inputs (quasi-fixed factors) then there exists a "variable" cost function: (14) In CV = H( n Y.,..., In Y, In P*,..., In P 1 m 1 n-q ' rt Z,..., n Z, t), n-q where CV = Z P.X. and the Z represent quasi-fixed factors, i.e., i«l * * * case. See Ohta (1974) for discussion of (13) in the single output

14 the subset of the X, which are not necessarily in long-run equilibrium. The variable cost function (14), like the total cost function, provides all the information required to infer the structure of production. In particular, comparative statics analysis implies the following relationships between the derivatives of (1) and (14): (15) 3 In CV/3 In IL * - FV./EF., l Yi Xi (16) 3 In CV/3 In z. = - F_./ FY,, X Zil Al (17) 3 In CV/3t = - F./EFV. t a1 Combining these relationships with our definition of returns to scale and our two definitions of productivity growth yields: (18) r - (1 - E(3 In CV/3 In Z ))/Z(3 In CV/3 In Y.), (19) (20) d In X, Ü d In Y dt d ny. = 0 l d In X. = 0 l 3 In CV /f s 1 3t d In z 3 In cv / 3 n CV 3t 3 Ù Y The formulas for productivity growth from the total cost function and the variable cost function coincide when the firm See Lau (1976), who treats both (8) and (12) as normalized restricted profit functions.

15 minimizes total cost. This can be demonstrated by noting that minimization of total cost requires: (21) 3 In CV/3 In Z± - - P±Z±/CV. It follows that: 9 In CV /. - 9 In CV V1 _ CV aj^i_gv. 3 ^ CT (22) gj ^1 - X d in z±) ' CT at at III. Estimation of the Structure of Production and Productivity Growth in U.S. Railroads Using the Variable Cost Function Our sample consists of cross sections of Class I railroads for 1955, 1963, and 1974, with 58, 56, and 40 firms, respectively. There are four output indexes: (i) ton-miles of freight, (ii) average length of freight haul, (iii) passenger-miles, and (iv) average length of passenger trips. There are three variable input indexes: (i) labor, (ii) fuel, and (iii) equipment (and materials). We specify tha,t way and structures capital is a quasifixed factor in the variable cost function. Estimates of the stock of way and structures for the individual railroads are available from the Interstate Commerce Commission. We employ the generalized translog multiproduct- cost function, proposed by Caves, Christensen, and Tretheway (1980) to 7Stock estimates for the beginning of 1955 and 1963 are avail able i- the ICC's Elements of Value. These figures have been updated to 1974 by Nelson (1975).

16 10 represent the variable cost function. This generalization of the translog form involves using the Box-Cox metric rather than the 8 natural log metric for output levels. This permits the inclusion of firms with zero passenger output which is inadmissible for the translog form. Using Y to denote the Box-Cox transformation of output y., Y. = (y, - 1)/X, and K to denote the Box-Cox transformation of way and structures capital k, K - (k - 1)/A, we write the generalized translog variable cost function for railroads: 4 3 (23) In CV = a + Z a.y. + cue + Z 3. In P \ Z I 6..Y.Y. + h ô K.. ij i 3 KK + h Z Z y., tn p. In P. + Z Z p.,y. In P. ij 1J x ] ij ^ J + Z pv.k n P. + Z 9.K Y.. j Kj 3 j Kj J The estimation of (23) with pooled data from 1955, 1963, and 1974 would be undesirable because it would not allow for differences among the years in the structure of production. On the other hand, with data only for three widely-spaced years it would be undesirable to introduce the 'usual integer measure of time. We overcome this problem by introducing binary Q The translog multiproduct cost function is discussed in Burgess (1974) and Brown, Caves, and Christensen (1979).

17 11 variables for 1955 and 1974 (e.g. D_5 = 1 in 1955, zero elsewhere) which allow the structure of production to be different from that of These binary variables are allowed to interact with the output and way and structures indexes and the input prices. The variable cost function augmented by these binary variables can be written: (24) In CV = a + a.,dcc + ct-.d,, + ZY. (a. + a_,,d-, + a.,.d.,) o x l 55i 55 74i 74' + K(aK + a55kd55 + a74kd74) + I In P±(ßi + ß55.D55 + ß74.D74) + h % Wj + h 6kkk2 + * y y a u piln pj + EZ p..y. In P. + S p^.k In P. + Z6V.KY Without loss of generality we impose, symmetry on the Y.- and 6... For our application this results in (24) having 50 parameters to be estimated. We impose the theoretical requirement of homogeneity of degree one in input prices, using the following linear restrictions: (25) zb± = i, ze55i = o, zß74i = 0, Ey.. =0 v., p. + Zp.. = 0 V. j ij i Kj jkij i These restrictions reduce the number of parameters to be estimated from 50 to 39. Given the large number of parameters in (24) to

18 12 be estimated, it is desirable to make use of the following cost share equations implied by Shephardfs (1953) Lemma: (26) P.X. l i cv - Si = ßi + ß55iD55 + ß74iD74 + f±i ** Pj + E p±jy + p^k i = 1,..., n-q. We treat (24) and (26) as a multivariate regression system and proceed to obtain efficient estimates of the parameters using a modi- 9 fication of the technique proposed by Zellner (1962). parameters are presented in Table 1, along with their standard 10 errors. The theory of cost and production requires that the own-price elasticities of the variable factors be negative and that the The estimated 9 To avoid the problems created by singularity of the contemporaneous covariance matrix, we delete one of the share equations prior to carrying out the second stage of ZellnerTs technique. The resulting estimates are asymptotically equivalent to maximum likelihood estimates, as well as being invariant to which equation is deleted at the second stage. 10 The y. and P. were normalized such that the mean of the 1963 values is equal to unity. In estimating the parameters of (19) we considered various restrictions on the structure of cost. There were two hypotheses which could not be rejected when tested jointly: (1) The elasticities of cost with respect to the outputs and fixed factor are independent of. time, acc. = a..,. = acc - a_/tr = 0; and r 55i 74i 55K 74K * (2) the length of haul of one output does not affect the cost elasticity of the other output or the cost elasticity of the length of haul of the other output, 6«D = 6DU = ô-«=* 0 (subscripts as IK rn RH defined in Table 1). To estimate (19) we have used a value for X of 0.1. Caves, Christensen and Tretheway (1980) have provided evidence that estimates and standard errors of other parameters are generally insensitive to varying values of X around 0.1.

19 Table 1 ESTIMATED PARAMETERS FOR THE GENERALIZED TRANSLOG VARIABLE COST FUNCTION Coefficient Estimate Standard Error Coefficient Estimate Standard Error Coefficient Estimate Standard Error R STT JHH W ^H TP PR a. KK LL Y EE Y QQ Y LE LQ QE»TL 'te 'tq 'hl he DPL 3PE DPQ 3RL 3RE 5RQ 5L55 $L74 *E55 *E74 *Q55 lq74 'KL PKE PKQ \t QKH %? QKR Subscripts: T H P R K L E Q » ton-miles - length of haul - passenger-miles - length of trip - structures - labor - fuel equipment-materials

20 14 Hessian matrix, [3 CV/3P±3P.]t be negative semi-definite. We find that the parameter estimates in Table 1 satisfy these regularity conditions for every railroad in all three cross section years. In Table 2 we present the own-price elasticities and the Allen cross elasticities of substitution for the three variable factors, evaluated for the average firm. Evaluation of productivity growth, (19) and (20), requires estimates of the elasticities of variable cost with respect to the output indexes and the stock of way and structures. These five elasticities are presented for the average firm for each of the three cross section years in Table 3. The degree of scale economies can be computed from these elasticities using (18). These estimates are presented in Table 4 for the average firm in The average firm is defined as follows Passenger- Ton-Miles Average-Haul Miles (billions) (miles) (millions) Structures Average-Trip (millions (miles) 1963 $) < Input Prices Equipment Labor Fuel

21 15 Table 2 Own Price Elasticities and Elasticities of Substitution From the Estimated Variable Cost Function for U.S. Railroads (Evaluated at the Average Firm for Each Year) Own Price Elasticities Elasticities of Substitution Equipment- Equipment- Labor- Equipment Labor Fuel Labor Fuel Fuel , Table 3 Elasticities of Variable Cost Evaluated at the Average Firm in Each Cross Section Year Elasticity of Variable Cost with Respect to Ton Miles Average Freight Haul Passenger Miles Average Passenger Trip Structures

22 16 Table 4 Estimates of Returns to Scale from the Variable Cost Function (evaluated at the average firm each year) ,196 1, r- assumes that proportional increases in ton-miles and passenger-miles are due entirely to increases in average haul and trip length. r^, assumes no increases in average haul and trip length. Table 5 Average Annual Rate of Productivity Growth Estimated from the Variable Cost Function (growth rates in %/year) Shift in variable cost function Productivity growth outputs fixed Productivity growth inputs fixed (r, ) Productivity growth inputs fixed (r~)

23 17 each cross section year. These estimates, denoted r-, are based on equi-proportional increase of all four output indexes. This implies that increases in ton-miles and passenger-miles are entirely due to increased distance with fixed tonnage and numbers of passengers. The alternative extreme would be to define scale economies with ton-mile and passenger-mile increases resulting entirely from tonnage and passenger increases with distance fixed. This is accomplished by using only the ton-mile and passenger-mile cost elasticities in evaluating (18). These alternative estimates of the degree of scale economies are denoted by r in Table 4. Scale economies could also be defined to reflect any desired combination of increases in distance as well as increases in tonnage and passengers. In this case the degree of scale economies would fall between the two sets of estimates in Table 4. The first step in evaluating productivity, growth according to either equation (19) or (20) is to evaluate the shift in the variable cost function with respect to time, 3 tn CV/3t. Time does not enter our cost functions directly; it is represented by the binary variables D-s and D7^. Thus we measure the shift in the cost function between years t and s as: (27) - 8 ^ CV = Un CV(r) - In CV(s))/(r-s), where Zn CV(k) is the variable cost function evaluated with the binary variables set at appropriate values for year k.

24 18 In evaluating In CV(t) and In CV(s) all variables other than the binary variables are held fixed at the means of the values for the 12 representative firms for years t and s. The first row of Table 5 presents our estimate of the shifts in the cost function between each pair of years. We have presented these estimates as average annual percent growth rates by multiplying the results of (27) by 100. The remaining rows of Table 5 present measures of productivity growth. When productivity is defined according to constant output levels we obtain the estimated growth rates shown in row 2 of Table 5. In the case where inputs are held constant we compute productivity growth using the two alternative estimates of scale economies from Table 4. These estimates of productivity growth are shown in rows 3 and 4 of Table 5. The three alternative estimates all indicate that there was rapid productivity growth from 1955 to 1963, on the order of 3.5% to 4.0% per year. From 1963 to 1974 the rate slowed to less than 1% per year. Over the full period productivity growth averaged approximately 2% per year. 12 Our method for evaluating 9 In CV/3t, including the use of the means of the representative firms, can be viewed as an application of the quadratic approximation lemma to determine the portion of cost changes which are due to time. See Diewert (1976).

25 19 IV. Estimation of the Structure of Production and Productivity Growth in U.S. Railroads Using the Total Cost Function. If the firm minimizes cost with respect to all inputs, including structures, then the correct cost function to analyze the structure of production is the total cost function. Caves, Christensen, and Swanson (1980) have presented estimates of a total cost function for U.S. railroads based on the same data set which we employ. In this section we use their estimated total cost function to infer the structure of production and productivity growth in U.S. railroads. These inferences can then be compared to those from the variable cost function, presented in the previous section. We begin by comparing the input demand structure for the two estimated cost functions. The own price-elasticities and elasticities of substitution from the total cost function are shown in Table 6. These results are clearly not as satisfactory as those of the variable cost function shown in Table 2. The total cost function yields capital and labor own-price elasticities which are essentially zero. In fact, the point estimate of the capital price elasticity is positive, in violation of the requirements of our cost minimization model. In Table 7 we present the cost elasticities implied by the estimated total cost function. In Table 8 we show the returns to 13 By itself this slight violation of the regularity conditions is not a problem., Very small adjustments of certain of the cost function parameters would eliminate the problem and have no effect on the cost elasticities and productivity growth estimates which are our primary interest. See Caves, Christensen, and Swanson (1980),

26 20 Table 6 Own-Price Elasticities and Elasticities of Substitution rom the Estimated Total Cost Function for U.S. Railroads (evaluated for the average firm in each year) Own Price Elasticities Elasticities of Substitution Capital Labor Fuel Capital- Capital- Labor- Labor Fuel Fuel Table 7 Elasticities of Total Cost Evaluated at the Average Firm in Each Cross Section Year Elasticities of Total Cost with Respect to Average Passenger- Average Ton-Miles Freight Miles Passenger Haul

27 21 Table 8 Estimates of Returns to Scale from the Total Cost Function (evaluated for the average firm in each year) r r, assumes that proportional increases in ton-miles and passenger-miles are due entirely to increases in average haul and trip length. r«assumes no increases in average haul and trip length. Table 9 Estimated Average Annual Rate of Productivity Growth Using the Total Cost Function (growth rate in %/year) Productivity growth, outputs fixed Productivity growth, inputs fixed (1) Productivity growth, inputs fixed (2)

28 22 scale implied by these cost elasticity estimates. Despite the fact that the variable cost model yields more credible estimates of the input demand structure, both the variable and total cost models produce very similar estimates of scale economies. Both models show fairly strong scale economies if output increases come in the form of increased haul and trip lengths. Both models show only slight scale economies if output is increased with length of haul and trip held fixed. In Table 9 we show estimates of productivity growth from the total cost function. The estimates for the full period, , are considerably different from the estimates in Table 5 from the variable cost function. The figures for the subperiods reveal that most of this difference arises in the early subperiod: estimates of productivity growth are much higher for the variable than for the total cost function. In the more recent subperiod, , both cost functions yield small estimates of productivity growth; however the estimates from the total cost function are again below the estimates from the variable cost function. The total cost function we have used is taken from Caves, Christensen, and Swanson (1980), who used it to estimate industry average elasticities to construct index number measures of aggregate railroad productivity. Even though both their estimates and our estimates of productivity growth are based on the same total cost function they find considerably higher estimates of productivity

29 23 growth than we obtain in this paper. This difference is due to their aggregation to the industry level. The estimated total cost function implies little shift in the production structure, but strong scale economics. Much of the effect of these strong scale economies on cost is hidden in an application of index numbers to industry aggregate data. This is because the industry totals show little output growth and hence little effect of scale; whereas output of the average firm has grown substantially through mergers and consolidations. The scale effects associated with this.growth of firm size show up as productivity growth at the industry aggregate level. V. Concluding Remarks The measurement of productivity growth is an attempt to distinguish shifts in the structure of production from movements along the efficient production surface. In this paper we have developed definitions of productivity growth for more general structures of production than have previously been considered. In addition we have showed how to develop measures of productivity growth under a different set of behavioral specifications than have been used in other applications. We have applied our definitions and measurement procedures to data from the U.S. railroad industry for the post World War II period to obtain new measures of growth in U.S. railroad productivity.

30 23 growth than we obtain in this paper. This difference is due to their aggregation to the industry level. The estimated total cost function implies little shift in the production structure, but strong scale economics. Much of the effect of these strong scale economies on cost is hidden in an application of index numbers to industry aggregate data. This is because the industry totals show little output growth and hence little effect of scale; whereas output of the average firm has grown substantially through mergers and consolidations. The scale effects associated with this.growth of firm size show up as productivity growth at the industry aggregate level. V. Concluding Remarks The measurement of productivity growth is an attempt to distinguish shifts in the structure of production from movements along the efficient production surface. In this paper we have developed definitions of productivity growth for more general structures of production than have previously been considered. In addition we have showed how to develop measures of productivity growth under a different set of behavioral specifications than have been used in other applications. We have applied our definitions and measurement procedures to data from the U.S. railroad industry for the post World War II period to obtain new measures of growth in U.S. railroad productivity.

31 24 The measurement techniques which we utilize permit the production structure to have any degree of returns to scale. In addition we relax the assumption that the firm minimizes total c6st, and assume instead that the firm minimizes the cost of employing variable factors subject to predetermined levels of quasi-fixed factors. The estimation of productivity growth under these specifications requires econometric estimation rather than index number analysis. We utilize a variable cost function to estimate the structure of production and productivity growth for U.S. railroads. Our estimates of the variable cost model show that returns to scale are sizeable for U.S. railroads if output changes are accompanied by changes in haul and trip lengths. Since haul and trip lengths have changed substantially during the period we are examining, we conclude that it is important to permit non-constant returns to scale in the production structure. We compare the results from our variable cost function to those implied by the total cost function. Although the two cost functions imply similar estimates of returns to scale, they yield different estimates of productivity growth, especially in the early period. We conclude that the behavioral assumptions underlying cost function analysis have important implications for the measurement of productivity growth.

32 25 References W. J. Baumol (1975), "Scale Economies, Average Cost and the Profitability of Marginal Cost Pricing," in R. E- Grieson, ed., Essays in Urban Economics and Public Finance in Honor of William S. Vickrey, (Lexington, Mass., D.C. Heath, 1975). D. F. Burgess (1974), "A Cost Minimization Approach to Import Demand Equations," Review of Economics and Statistics. 61, 1974, pp R. S. Brown, D. W. Caves, and L. R. Christensen 1979, "Modelling the Structure of Cost and Production for Multiproduct Firms," Southern Economic Journal, 46/1, July, pp D. W. Caves, L. R. Christensen, and J. A. Swanson (1980), "Productivity in U.S. Railroads, ," Bell Journal of Economics, Spring, forthcoming. D. W. Caves, L, R. Christensen, and M. W. Tretheway (1980), "Flexible Cost Functions for Multiproduct Firms," Review of Economics and Statistics, forthcoming. W. E. Diewert (1976), "Exact and Superlative Index Numbers," Journal of Econometrics, 4, 1976, Interstate Commerce Commission, Bureau of Accounts, "Elements of Value of Property Used in Common Carrier Service," unpublished working papers, annual. D. W. Jorgenson and Z. Griliches (1967), "The Explanation of Productivity Chang-." Review of Economic Studies, 34, 1967, pp L. J. Lau (1976), "A Characterization of the Normalized, Restricted Profit Function," Journal of Economic Theory, 12, 1976, pp

33 26 D. M, McFadden (1978), "Cost, Revenue, and Profit Functions," in M. Fuss and D. M. McFadden, eds., Production Economies: A Dual Approach to Theory and Applications, (Amsterdam: North-Holland), R. A. Nelson (1975), "Values of U.S. Class I Railroads, Market, Book, and Net Reproduction Values Estimated for December 31, 1973," Association of American Railroads, Staff Studies Group, Consultant Report M. Ohta (1974), "A Note on the Duality Between Production and Cost Function: Rate of Returns of Scale and Rate of Technical Progress," Economic Studies Quarterly, 25, 1974, pp J. C. Panzar, and R. D. Willig (1977), "Economies of Scale in Multi- Output Production," Quarterly Journal of Economics, 1977, pp P. A. Samuelson (1947), Foundations of Economic Analysis (Cambridge, Mass.: Harvard University Press). R. W. Shephard (1953), Cost and Production Functions, Princeton, Princeton University Press. Rt A. Solow (1957), "Technical Change and the Aggregate Production Function," Review of Economics and Statistics, 39, pp A. Zellner (1962), "An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias," Journal of the American Statistical Association, 58, December, pp