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Eilí'O (rffGi) ij«'[,"°a o S68I ISIEO 99Ô1 S95C0 reit líSS'O nosí SEE* U (9S/ÍS61) «nwJisnv t 9?Sl SSIE0 OOU 6E9I*0 p061* pOfrES'O I0SJS?6I> |w»l»2A«K 'C <>cse fliSIO /.Oil E9S10 íífiE 8«;oo C9IC OttľO (Iífill »>Fwj»j g aut Híl'O Hít EEEľO otet 61110 i«e> 99ÖT0 (KCl) sejmS PBWJI I ■»s .1/7 «s J/7 «IS -j/7 «« A íl Äjjwno^» o:c OCE JK IK míiorfsní^ ŕ>"" {Äí«rt^jujir jrfwj^) sffVi'H' pug puo tiijiitpur {'tvoi) ssjotf ^"Ul f*" "í"dw/ -"xpry m> djí>(7 jouoiíoujsiuj 414 MA HC HK1II.0VB If the efficiency of (inn* vane« neutrally,1 ah indicated by (Jul «rrnr term in [I), iiiľl Mt" price« paid lor FiKiloľH vury J -« !■■ firm tu linn, then the levels of input arc mitdetormiucd IniUipnmliinUy but are determined jointly by the firm's efficiency, level of output, ami Ml« factor price.« it mnut pay. In short, n litted relationship between inputs aud output fa a tostflaeitt relation tlmi docs Hot describe the production function nt. all but only tho not effects of differences among firms. (Kor n more general diseiiřwiou, BOO [KÍ, Iß.|.) In such oases, however, it may be possible to (it the reduced form of a system of structural relations such as (I) and (3) and to derive estimates of the structural parameters from estimates of the reduced-form parameters. Net oidy does it turn out ti> be possible in this case, but an important reduced form turns out to be the cost fuuetion: where ft =» ria^'aiH,^)-"', anil Or + j. pp. 170-7(1. Most generali v( it .seems lo have been handled by deflating tost figures by an index of factor price», o procedure that Johnston ]10] shows typically lea/Is to biaa in the estimation of the cost curve unless correct weights, which depend on (unknown) parameters of the production function, are used. It seems strange that no one has taken the obvious step of including factor jiricPA directly in Um m*t Juiirliini If price data »re available for the I A moUül inc!ii|ii>Kii>ri« IHMI-IWetnU Variation* in cllldt'iii-.y at bhu foini „ •? (n.iij.-i-i.^"!*^"» W im iU«:iiBA!il in my papur "On Mcoauiwiiwnt of Relative Eľcinunc liftlciuncy," uJisUust, Reon/mttrica, 28 (July I 1X10). »505. It is intonating to note that rliwpila the rumple» w,\.y in which ÜW ttUMtotli vliuiools »4, >',- and "i ontor, tlumí nrv eirnmiwUn»»-* timW «high it in potwihln U> <*tini«to tin» paiinnnioni in wioh a produclimi fhiifltin». II ' -.v.- this puinb to i I n-.-i .um Lfxawa. It i-i true, of course, only if atl linns hav« Lho nnme production function, except perhaps for ditferenees in the constant term, so that aggregation diffumlties mar be nn^Wted. Kttunu lo Seat» in ICleriricity Supply 41 collaruction of an index and price* do lint move pilipniti.....d.-lv. m .vliieh ctutn no bins would result limn dellntioii, why not use the r.xlm information aflontodl What form uf production function in appropriate for electric power* The generalized Cobb.Douglas function presented above is attractive for two reasons: First, it leads to a cost function that is linear in the logarithms of the variables #) 0 = £ +1 y + % /', + ^ j>% + 23 pt + p. r r r r where capital letters denote logarithms of the corresponding lower-case letters. The linearity of (5) makes it especially easy to estimate. Second, a single estimate of returns t« scale is possible (it is the reciprocal of the coefficient of tJie logarithm of output), and returns to scale do not depend on output or factor prices. (The last-mentioned advantage turns out to be a defect as we shall see when we come to examine a few statistical results.) But does such a function accurately characterise the conditions of production in the electric power industry? A casual examination of trade publications suggests that once a plant is built, fixed proportions are mere nearly the rule. Support for this view is given by Koiiiiy» 11 l |. who found that data on inputs and output for individual plants were botler approximated by a fixed-proportion»« model that allowed differences in the proportions due to Ronle. A simplified version of Kimiiyu's model i*a J-, «iff*1. (6> - - = "a!/1'*. M the linn level, however, then1 are ninny |KMtHÍI)Ulti(W Tor milwlilulion that may go unnoticed at the plant level; for example, labor and fuel may bo Substituted for capital by using older, less efficient plants more intensively nr by iming a large tunnlH-r of .small plants rather than n few large ones. Given persistent differences in the factor prices paid by different linn«, cross-section data should reflect such possibilities of substitution. Certainly, as a provisional hypothesis, a gv,iicr»li'/>c-d (!ubb-l)ouykis function may bti ap p rep r i ate, ■ŕthlflll g i» iiXUtSMintlH, ti. uiHilil ki ii|i|ir-.|irinli. In ratfanoto Ihu pOoflraMuitfl m )(.| l>y IoiihI n.^unnat. An «lijmiliiin to clitM. ln.wcvei, 1« tlw fact Ihul, \t iiKlividtinl plant« aio <:iiti«idcred,Uio output allocated Uieaek »not exogenous: w-e Wrstfiold (ftfl.pp, 15-S1J. Ptertlicrmnre. Komijft tl nxngonoiBi in i» uruai strati.....My nbjidiou how i« rl*»«i|y k-J.iU-.I I« the one ini'-.-l by Hushed (f«ec p. 169): however, while the endogenioity of output at the plant level Ír olrar, ilN ondogoníflitý at the firm Ipvi-1 for a member of a jmwpr pool is conjee turu). an MA1W MKKTXtVR iL would, of course, be preferable to lt*t whether «igniliennt milutitulioii among factors occurs at the firm level. The use of the generalized Cobb-Douglas unfortunately Hoes not permit us to do so except in a very general way, silice its form implies that the elasticity of substitution between any pair of factors is one. A more general form, which has both the Cobb-Douglas and fixed coefficients as limiting cases, has recently been suggested by Arrow, Minnas, Chenery, and Solow [I]. Constant returns to scale are assumed, but the form can he easily generalized; in a more general form it is In this caso returns to scale aro given by the rutin bjf and the elasticity of substitution between any pair of factors cau be shown to be 1/(1 —6). In the special case in which 6 =/ it can bo shown that the limiting form of (7) as the elasticity of substitution goes to zero is ________fi________ ________H_________________gn \ (a, + „3 4. a.y- - 1 ' (a, + «3 + atf* - I ' <*, -\- a3 -i- «$** - Ij' (8) or fixed coefficients, and the limiting form as the elasticity of substitution goes to one is (9) y = (a, + a, + o|)*^'W«^.»^tai+*!*^/l*»+«ř*-.Í( or Cobb-Douglas. Although I have not formally demonstrated the fact, it is possible that the limiting form of the more general ease (7) is something like the Koiuiya model as the elasticity of substitution tends to zero, and like the generalized Cobb-Douglas as it tends to one. Unfortunately, in its generalized form (7) is quite difficult to estimate from the data available. Furthermore, although clearly superior to the generalized Cobb-Douglas form, (7) still implies that the elasticity of substitution between any pair of factors {o.g., labor capital and fuel capital) is the same, which hanily seems reasonable, Other generalisations are possible, but none that I have found thus far offers much hope of being amenable to a reasonable estimation procedure. If the generalized Cobb-Douglas form is adopted, however, relatively simple estimation procedures can be devised for evaluating the parameters of the production function. The reduced form of (1) and (3) that incorporates all but ono of the restrictions on the parameters in the derived demand equations (which arc the more usual reduced form) is nothing but the cost function. The only restriction not incorporated in (4) or (5) is that the coefficients of the prices must add up to one. It is a simple matter te incorporate this restriction, however, by dividing costs and two of the prices by the remaining price (it doesn't matter either economically or statistically which price we choose). When fuel price is used as the divisor, the result is y = nun l/rt-nm* '•> Smi« in Kfrrtrú-ilt, fíupfOfj 417 do) fl_r,=jř+l F+ä<*y-*<$+ä£P.-P(#+ v- which will he called Model A. Model A assumes that we have relevant data on the "price" of capital and that this price varies significantly from firm to firm. If neither is the ease, we are in trouble. Most of the results presented here are based on Model A, but the data used for this price of capital arc clearly inadequate. (See Appendix B.) If one supposes, however, that the price of capital is the same for all firms, which is not implausible, one can do without data on capital price and use the restriction on the coefficients of output and prices to estimate the elasticity of output with respect to capital input. The assumption that capital price is the same for all firms implies (II) ,.•„*•* |.I Y i-tlp, fSSp.+ K, r r r where K' = K -i (a,lr)Pt. since the exponents »f the input levels in (I) are assumed to Im> the samo for all firms. Kquation (11) is callod Model B. 2. Some Statistical Results akd Their Isterpretatiou Estimation of Model A from a cross section of firms requires that we obtain data OB production cost«, total physical output, and the prices of labor, capital, and fuel for each firm; for Model B »c do not need the price of capital, since it is assumed to be the same for all linns. Details of the construction of these data for a sample of 145 privately owned utilities in 1055 are given in Appendix B and are not discussed here at any length. Suffice it to say that these data are far from adequate for the purpose, and I now believe that a better job could have been done with other Sources. The results from the least-squares regression suggested by equation (10) are given in line I of Table 3; the interpretation of these results in terms of the parameters of the production function is given in line I of Table 4. The It2 is I) 93, which is somewhat unusual for such a large number of observations; increasing returns to scale are indicated, and the elasticities of output with respect t*i labor and fuel havo the right sign and are of plausible magnitude; however, the elasticity of output with respect to capital price has the wrong sign (fortunately, it is statistically insignificant). The difficulties with capita! may be due in part to the difficulty I encoun-li-ri'il in measuring both capital costs and the price of capital. The former were measured as depreciation charges plus the proportion of interest on long-term debt attributable to the production plant; the figure for capital price was compounded of the yield on the firm's long-term debt and an index of construction costs. Depreciation figures reflect past prices and purchases of capital equipment, whereas the price of capital as I constructed 63 420 MÄllO HRMbOVn Ta III« f\ licmiUä fn>m K'-jrcxi«»» Ht,*M oii M» < f offfirirui Itrytrt&tim No. y r, i; h' V ,rif.il', HhVV1;i 4 ■ I Tapi.*; 0 Utlum» i- Ne.nl* taul KlmtimtiM of Output müh Jtt'jwri i« yarfrmR luf/Hl* J)rrÍľ.'tfi<'fľ Itrtuti* t'lfrutf'l >M Till'tr B -fttr 145 pirmitiu IBQff luffuumiuH iVo. /ŕri'mi* ta tSnUr. 1,38 ťUtnti'itg •>/ íl»/)."' ■»'»"' /.V-/"' t« biliar (iapiiai Vttit 0.67 0.03 0.69 VIA 2.77 0.69 1.311 0.74 VI« 1.51 -0.62 n.nit 0.33 VIo 1.03 -0.01 0.27 0.46 Vli. 1.08 0.35 -0.34 0.82 Vis 0.97 0.68 0.03 0.88 the estimate of returns to scale if they are decreasing.6 But elsewhere Friodmnn holds that a uniform distribution is not likely to occur; in fact ho says, "The firms with the largest output are unlikely to be producing at an unusually low level; on the average they are likely to be producing at on unusually high level; and conversely for r.hoso that have the lowest output" {[UL p. 237). "This »rgument io*f* through the might. Log output Figure 1 the form of ihn funotiou that conBlruiwe it W 422 MAMO NJSKM>VB STC, Output ricuat 2 The situation described by Friedman is piotured in Figure 2 by the shaded areas A, B, and C, which refer, respectively, fco observations on firms ivil.li unusually low, usual, and unusually high outputs. The Friedman explanation does produco a residual pattern similar to that observed. Regroaaion H, Table 3, is designed to test this explanation) for Model A. A corresponding teat for Model B vos not made. Since "usual" output cannot be directly observed, the hypothesis was modified slightly by identifying departure from the usual with large changes in output from the previous year, the assumption being that firms with stable output were likely to bo near the optimal long-run output.* Thus, the absolute percentage changes in output should be positively related to total costs. Unfortunately, they are negatively related and significantly so. Part of the explanation for this unexpected result is suggested by a more careful examination of the data. Almost all firms with large changes • Capacity flßtires mightIwvp liro» nana. nawtsmr, bhoso avaibblo appear i" bo boiuo-wlinl unrealistic. Those aio basod en gcnemior nmnc-pl»tc ratings, which renn to Hie maximum output that can bo produced without overheating. According to the Kcdcml Power Commission, however, unit« of the some size, general design, and actual capability may show ns much a.» a 20 per cont dilTeronco in rating (|5J, p. xij- Furthermore, in a multiple-plant firm, total generator capacity a not the only (actor to be considered. Such defect* in the capacity figures also led to grouping firms by output rather than by capacity in the analyses of covarinnce presented below. Upturns t» Scale in Elaariciiy Supply 423 had positive changes and hnd been experiencing rapid growth for some time. It is well known, though unfortunately not iahen into account in these analyses, that there is b steady rate of technological progress in generating equipment. Sine*' expanding firms purchase new equipment in the process, the average age of a planí m those firms experiencing largo changes i" output is lower than that of (inns with more stable outputs. Hence, the former tend to have lower coats because of the inadequacy of the enpitaj-cost data to reflect, nhsolofwtaion.1 Tim*, while one would nut want to reject the Friedman hypothesis on the basis of this evidence, it clearly does not explain the residual pattern. 2. Fortunately, the observed result can be Oxplrtined by a much siinplur hypothesis, namely, that the degree of returns to scale is not independent of output, but varies inversely with it. Figure 3 illustrates this explanation: The solid line gives the traditional form of the total cost function, which shows increasing returns al. low output« and decreasing returns at high outputs. If we try to fit a function for which returns to scale are independent of the level of output, e.g., one linear in logarithms, a curve such as the dashed one will be obtained. The shaded areas A and B show the output ranges, high and low, for which total costs are underestimated. If the true cost function is not linear in logarithms, we can either lit an over-all function that reflects this fact or attempt to approximate the actual function by a series of segments of functions linear in logarithms Because of fitting difficulties and the problem of determining the form in which factor prices enter the cost function. I initially chose the latter course. Firms, arrttyed in order of asconding output, «ere divided into 5 groups containing 29 observations each. A list of the firms used in the analysts appears in Appendix C. The results of fitting five separate regressions of the form indicated by Model A are given inline.- 111a through IIIE of Table 3 nnd the corresponding implications for the parameters in the production function in lines 111a through Illr of Table 4. Similar results for regressions of Ihe form indicated by Model B are presented in lines YIa through Vis of Table* 5 and 6. The results of these regressions with respect to returns to scale are appealing: Except for statistically insignificant reversal between groups c and n, returns to scale diminish steadily, falling from a high of better than 2.5 to a low of slightly less than I. which indicates increasing return? at- a diminishing rate for all except the largest firms in the sample. However, in the case of regressions III. the elasticity of output with respect to capital ' TnľiKniKiit of capital I'OgUns the tOUťeo of on« of I he ncml sonoue eliurlccunin^v "flhe j mnout, study, n.-, indeed capital measurement i» in most stadieaoF production, fitilnw'a ■■.■cont couliitnitiuii to the study of Ihr ngetťgntc production funrlnm [ IS] «flrrs con-»itii-rnlilc prom UK) of an appropriate measure of capital used in Ihp production of riwt*ic power, I hopo, in future work, to miikť aso uf a model of protUiotfon lhát involves fixed conllifiienM e-r ;«**' at tlio plimt level, but tliut permits cotxtibittuu Of input* mul that changes over time ee ante. f | J* S M | £ Q - g i 2 3 u I! H g í J'S s s t i! Ľ sill íl :■' a -* tí 81 fill ■£2. c * Ulli 11 a •= S ,2 — K -= 5 1 unii! £ 5 I« 5í ?£ slil 01 T3 '. íl i. -c C !B i -^ = e -= £ M O Zl íil ľ' Q ■ 5 K Níl a : = 5 O - r. z "a v -= ja f] AH! :-i i^ § r^ 'li H B ttíHiíí' r t Ľ * S -3 =-« .= 19 = "s — = j?J iil^äseä * í i i = « ■* ; - ; Im 1 ! ílíi»«« S 4 11 »H — >c t: 5% I; ■»■g-a i «j -n i) — _ľ 3 - „ 3 íl i« íl fi 5 * f j 3. S £ 'P .M Sff.fi 2. u S Jí P .« 5p — o -i *s — — *-í — J! í S-I .E .S ft, t, «Ti»- íl : í* tí II c = ;; II tO — ES» « f! C 9 «1 lif s t ll t/ _ n « — .2 ^1 «i; i ŕ| ill :j =■ ~ n j p 'ä I! ■I ä § ■=! ¥5 si Hfl íl» i? iu>< it! í í fi -; - K 1*>0 »fill 1 P- | íl S 11 n U ?i B i- -J íl i* m -i * 2 Fľ« W> í les íl ■a ■U C JS •k 'č tí a H &. Ľ — '3 * -= ??-= 3 £ -1 I« S 1 > - "t £ Bil 1 H 8*3 M-§i l Š.S s: -ií-= -= ľ TJ -í S II I s s * íl If II i ■S -ä = d — = ^ u .SP C Q M a« s. -= .3 c S.* S 4 fit ž 1 «-i 420 MAltr NRltl^JVIt Thus, instead itf regnwinns «t tli« I'm'!» suggested by (10) (»' (II). WO lil. (U) c-p,- /r-f-*K + i?r, + -|/'. ■lU+^P'i-AI + r (Model Cí and (15) C - A" -(- a T |- /í Ks + ^ 7», + 2S P3 I K r r (Model D). Tli« results obtained for regressions based on Model Q and Model I) are reported in Tablo 7 fur regressions VI1 and VIII, rcapooUvoly, The impli-cations of those results for the production function aru given in Tablo 8. Notu that returns to scale and the other parameters hare been computed at live output levels only, bo that the results in Table 8 may be readily compared witb those in Tables 4 »mi 0. Perbajw tin- most sinking result nf the assumption of continuously and neutrally variable returns to scale of the form Suggested in (13) is the .substantial increase ill our estimate of the degree of returns 1«» scale for linns in the three largest size groups. Whereat« before. »'" found nearly constant returns to scale, it now appears that they are Increasing." In addition, all Taulk -. Results from Reuression» Based o« Modrín C and D for 14f> Firm* in 1055; Cottiinuotis Kcutral Variations in lielurns to Scale Mo-MC Tiep«suUfd Variable Wax f." - í'a ilow&sion jVo. Coéälcieiti K* VII y v p, - P% P, -P, i>.ir.i O.II7 11.498 (±o.oi2i uo.iui) 0.0B2 (±0-1511 0.11.18 Morfeí I>: liepemUtü Variable. Was (! Rtgrrssion So. Ooej/ifient R> vin Y y z*, ''. 0.131 (£0.064) 0.118 0.279 (10.013) (±0.224) 0.ÍS5 (i-0.054) 0.952 Ŕgvro) in parentheses ore ihn standard errors «f the coefficient"». " UsiiiĽ the varianoe-covarionoe matrix for th(* co*flmionlo in (14) or (15), one could OMtly compute, fór o given y, a condition»! standard error fur Iff, which could limn l*i Itthtrns f« Äífllfl Mt Klrelririf,, Supply jo, Tahuc h Jtcturna to Scale, urul /Haslicitits of Output with lies-pcct tu Various 'num.' ifrritvitfrom liesid/i rtaaenttdin Table 1 f or MS Firm/tin 1955 Hegrestio" Vil (MoJel 0) Itthinis to S. <,!.• Elitrtwiiy nf Output tritlt IttSjWrt to" fTiWMfi 1 nihil} t 'iifiiiiil ľu.t A •:.'.>•• 1.45 (US 1.20 K 2.21 1.12 0.14 U.»S C i.m 0.98 0.12 0.87 D 1.84 0.!)2 0.11 O.HI K l.f>9 «1.K4 11.11) 0.75 HqpvMaio,, VIII (JVouW/iJ ICilnrii/ to Scale* lilastiiit}/ of Output with Hespett to" Group Labor Capital Fuel A 3.03 0.85 1.41 0.77 II 2.30 0.(14 1.07 (».5» 0 2.01 U.ÖI! 0.U4 0.51 D 1.88 0.52 0.88 0.48 i-: 1.78 0.4« II.SO ».44 Evaluated at the median output for rach proiiji. the coefficients in both analyses arc of the right sign, and the results based mi Model II yield result» of plausible magnitude for the eltuUioity of output with respect to capital as compared with the elasticities with respect to labor and fuel. Analyses of oovarianoo, comparing regressions VII end I »uxid to t*et »hethor l/r woro BignifieeiiSly lesa lliaii one (i.e., "hcllirf ihc fording of ionmnalng rrhiniH wM 8f hi ial.iwüly tttfiiiinViitit). t"nn irtu lintel y. t. ho rngrenBiwi pr'flram used did liOtprint out the inverse ol (he inoimuil matrix, mj this test could not be mini«;. But thorn is little doubt, in view of the extremely mnnll «tiuirferd error« of th<- r*timau bn Elatiiticnllv nmíBtáni, IS .- h - Z/ X - í - i "81 U aa alls lílíi o p. ä-3 e i^ 11 li* s* j a If k í. ífffl 9 "3 S íl 5 1 ŕP* s 3 d ■S => Z H - 5 5 "í y B <*. U ti. Ä-tí 3 „a ?> ~ « = H « H3& Í4 ŕ 4 TT a > si JE '■<- t ['S > ■2- í*. ~- ~ P s , ä S <: ^ m ill Mi III •-If II ifjjj = :^í3í o — — -J — — 2 c = II ll = = J-i SI í i n p 11 í-JÍ? = í;í = £3 >11 š=í •* *Ä 3 "i O tí í J a. 3 .2 g í! I I i 9 'S ! S n J Ž' i-IJ I = y lil fit ä $ Ľ ap •-g I í ? 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