Economies of Scale: Replicating Christensen and Greene (1976) by Arianto A. Patunru Department of Economics, University of Indonesia 2004

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1 Economes of Scale: Replcatng Chrstensen and Greene (976) b Aranto A. Patunru Department of Economcs, Unverst of Indonesa 4. Introducton Ths exercse s based on a class assgnment n Unverst of Illnos, nstructed b Prof. Carl Nelson. It s to replcate some of the results on economes of scale reported b Chrstensen and Greene (Economes of Scale n U.S. Electrc Power Generaton, Journal of Poltcal Econom, 84:4, Part, 976). Chrstensen and Green updated the data used b Nerlove (955; Returns to Scale n Electrct Suppl, reprnted n C.F. Chrst, ed., Measurement n Economcs: Studes n Honor of Yehuda Grunfeld, Stanford Unv. Press; 963:67-98) to 97. The found that b 97 the bulk of electrct generaton n the Unted States came from frms operatng ver near the bottom of ther average cost curves. The concluded that a small number of extremel large frms are not requred for effcent producton and that polces promotng competton n electrc power generaton cannot be faulted n terms of sacrfcng economes-of-scale.. Replcatng Nerlove (955) Frst, we replcate some of the prncpal returns-to-scale results reported b Nerlove (955). The equaton estmated b Nerlove s: ln(cp3) β + β ln(kwh) + β ln(p3) + β ln(p3) where ln s natural logarthm and: cp3 kwh p3 p3 relatve total costs klowatt-hours as measure of outputs labor relatve prce wth respect to fuel prce captal relatve prce wth respect to fuel prce Nerlove (963) reports parameter estmates for β, β, and β as.7(.75),.56(.98), and.3(.9), respectvel, where fgures n parentheses are standard errors; and R of.93.

2 .. Estmaton Our replcaton elds: Source SS df MS Number of obs F( 3, 4) Model Prob > F. Resdual R-squared Adj R-squared.93 Total Root MSE.3976 lncp3 Coef. Std. Err. t P> t [95% Conf. Interval] lnkwh lnp lnp _cons Therefore we cannot replcate Nerlove s results precsel. Both estmates and ther standard errors are dfferent from what Nerlove obtans. However, we obtan R that s close to Nerlove s... Returns-to-Scale From the estmaton above we obtan a 95% confdence nterval for βˆ [.686,.755] βˆ as follows: Ths mples that, at the 5% sgnfcance level, there s no constant returns-to-scale, snce βˆ les outsde the nterval. Another wa to examne constant returns-to-scale s b testng the hpothess of β aganst the alternatve, β. The test statstc for ths s Comparng ths to crtcal value of Z ±.96, α. 5 leads us to rejectng the null hpothess. Agan, α we conclude there s no constant returns-to-scale. We can also calculate ths manuall usng the formula: CI [ ˆ ˆ β ± Z ˆ ] β α σ β.5 The formula s ˆ β σ β ˆ ; n addton we could also use F-test provded b STATA (see appendx), where we get F(,4) wth Prob > F.. Takng square root of ths F-stat gves us the t-stat above.

3 Now we are nterested to know whether the returns-to-scale s actuall ncreasng or decreasng. We use a pont estmate r 3 wth relaton to β as β r. Solvng for r we have: r ˆ β. 39. B the propert of Cobb-Douglas functon, we conclude an ncreasng returns-to-scale. Lastl, we can obtan postve economes of scale: e r Factor Share The demands for each factor of producton wll be postve onl f the factor share, α s postve,,, 3. The factor shares are determned b α β β ˆ β r. Therefore we obtan α x Usng ths result we can test the hpothess that α. Ths gves a test statstc of Thus, we cannot reject the hpothess. We conclude that the factor share of captal s not sgnfcantl dfferent from zero. Ths result s strkng snce t mples that captal s not beng used as a producton factor n electrct generaton. Ths mght be the result wh Nerlove was unsatsfed..4. Resduals In the followng fgure we plot the resduals from the estmated regresson aganst the output:.89.5 Resduals lnkwh 3 r beng the sum of factors exponents n a Cobb-Douglas functon (.e. returns-to-scale parameter). 4 The formula s α, where σ α var( ˆ β ) r σ ˆ β σ α r.65 ; and p-value s

4 In the fgure we see that the resduals are postve at small levels of output, negatve at medum levels, and postve agan at larger level of outputs. Ths U-shaped pattern s close to what Nerlove found and t means that the regresson results overestmate the true costs at ver low and ver hgh levels of output, and underestmate them at ntermedate levels of output. Fnall we note that the sample correlaton between the resduals and the output s zero. Ths s because n OLS estmaton, the explanator varables (here ncludng the output) are orthogonal to the error term,.e. the are ndependent. 3. Fxng the Data Next we use UPDATE data provded b Berndt (The Practce of Econometrcs: Classc and Contemporar, Addson-Wesle Publshng, 99). The data conssts of 9 columns wth 99 observatons. The columns are OBSNO (observaton number), COST7 (total costs), WH7 (output), PL7 (labor prce), P7 (captal prce), and PF7 (fuel prce). After examnng the data we suspect that there s a decmal placng error n PL7 column. Summar statstcs shows that the mean and standard devaton of PL7 are extremel hgh compared to those of P7 and PF7: Varable Obs Mean Std. Dev. Mn Max pl pk pf To fx ths, we dvde the PL7 entres b. We then obtan the followng statstcs: Varable Obs Mean Std. Dev. Mn Max pl pk pf whch s more comparable. 4. Cobb-Douglas Cost Functon After transformng the data nto relatve prces wth respect to fuel prce we construct a Cobb-Douglas cost functon as follows: ( COST / PF) βy β L β ( WH ) ( PL / PF) ( P / PF) β or, n ther log forms: ln c β + β Y ln + β L ln pl + β ln pk 4

5 where c : COST/PF : WH pl : PL/PF pk : P/PF and ln means natural logarthm. We obtan the followng results from estmaton: Source SS df MS Number of obs F( 3, 95) Model Prob > F. Resdual R-squared Adj R-squared.9778 Total Root MSE.99 lnc Coef. Std. Err. t P> t [95% Conf. Interval] ln lnpl lnpk _cons The table shows that both ln pl and ln pk (.e. ln(pl/pf) and ln(p/pf), resp.) are not sgnfcant at the 5 percent level,.e. the t-statstcs are.33 and.68, respectvel, wth p-value of.86 and.96. We can calculate the derved value of the coeffcent for ln(pf) as follows: ln(cost) ln(pf) β + β Y ln(wh) + β L [ln(pl) ln(pf)] + β [ln(p) ln(pf)] ln(cost) β + β Y ln(wh) + β L ln(pl) + β ln(p) + ( - β L - β ) ln(pf) Thus, the coeffcent for ln(pf) s ˆ β F ˆ β ˆ β L The standard error s calculated as follows: Var ( - βˆ L - βˆ ) Var (-( βˆ + L Var ( βˆ L + βˆ ) βˆ )) Var ( βˆ L ) + Var ( βˆ ) + Cov ( βˆ L, βˆ ) *(-.7568).6363 se ( ˆ β ) F Next, followng Berndt (99) we calculate the returns-to-scale as follows: 5

6 r ln c ln ˆ β Y.63 Ths result mples an ncreasng returns-to-scale. In addton, we have a postve economes-of-scale (e r.). 5. Augmented Nerlove Model To allow the returns-to-scale to var wth the output, we add the quadratc output term nto the estmaton. That s, we estmate: ln c β + β Y ln + β ln sq + β L ln pl + β ln pk where lnsq : [ln(wh)] (ln ) The estmaton elds: Source SS df MS Number of obs F( 4, 94) Model Prob > F. Resdual R-squared Adj R-squared.996 Total Root MSE.3758 lnc Coef. Std. Err. t P> t [95% Conf. Interval] ln lnsq lnpl lnpk _cons The result shows that onl ln pk s not sgnfcant at the 5 percent level,.e. wth t-statstc.78 and p-value.435. Meanwhle, the coeffcent for the quadratc output term s sgnfcantl dfferent from zero wth t-statstc.38 and p-value.. 5 Snce WH s not dvded b PF, then the derved coeffcent for ln(pf) and ts standard devaton can be calculated wth the same formula as n Cobb-Douglas, that s, ˆ β ˆ β ˆ β.6559 wth se ( ˆ β ) F L Now the returns-to-scale s calculated as: F r ln c ln ˆ β + * ˆ β Y ln 5 Ths s the same as testng for H : β usng F-stat (see appendx) that gves F(,94) 9.4 wth Prob > F., meanng that the null hpothess s rejected. 6

7 Therefore, the returns-to-scale vares wth the output level. We can plot ths mpled returns-to-scale as a functon of output as n the followng graph: Fgure. Returns-to-Scale and Output, Augmented Nerlove Model.64.5 rtsquad output Note: rtsquad: returns-to-scale for model wth quadratc output term. The graph mples that substantal scale economes are obtaned at low levels of output. Ths s consstent wth Chrstensen and Greene s fndng. 6. Chrstensen and Greene s Model A Fnall we estmate the cost functon usng Chrstensen and Greenes Model A. The used the followng translog cost functon: where nputs 6. lnc α + αy lny + γ (lny ) + j j α ln P + γ j ln P Pj + γ Y j lny ln P γ γ, C s total cost, Y s output, and the P s are the prces of the factor 6 The relatonshps among the parameters are such that the cost functon s homogenous of degree one n prces. That s, α, γ Y, γ γ j γ j j j j 7

8 For our estmaton we adjust ths functon nto: ln c β + + β Y ln + β (ln ) + β L ln pl + β LL (ln pl) + β ln pk β (ln pk) + β (ln pl)(ln pk) + β (ln pl)(ln ) + β L We obtan the estmaton result as follows: LY Y (ln pk)(ln ) Source SS df MS Number of obs F( 9, 89) 4.36 Model Prob > F. Resdual R-squared Adj R-squared.99 Total Root MSE.336 lnc Coef. Std. Err. t P> t [95% Conf. Interval] ln lnsq lnpl lnplsq lnpk lnpksq lnplpk lnpl lnpk _cons It s shown that ln pl, (ln pl), (ln pl)(ln pk), and (ln pl)(ln ), that s, all ln pl and ts nteracton terms, are not sgnfcant at the 5 percent level. The t-statstcs for these varables are.38,.7, -.95, and.5, respectvel, wth p-values of.75,.46,.55, and.6. The translog cost functon used b Chrstensen and Greene does not constran the structure of producton to be homothetc, nor does t mpose restrctons on the elastctes-of-substtuton. However, these can be tested statstcall. That s, homothetct restrcton s such that γ and homogenet restrcton s such that γ and γ. Y Y Takng ths nto our estmaton, we can examne the homothetct n output b testng whether or not the coeffcents of (ln pl)(ln ) and (ln pk)(ln ) are both zero. Our test gves an F-stat of 5.5 wth p-value of.84. Therefore, we reject the hpothess at the 5 percent level. In other words, the cost functon appled to 97 data does not exhbt homothetct. On the other hand, the assessment of homogenet n output nvolves the testng of whether the above restrctons hold (the coeffcents βˆ LY and βˆ Y are both zero) and the coeffcent of (ln ), βˆ s also zero. The test gves an F-stat of wth p-value of 8

9 .. So, we reject the hpothess for homogenet, too. Ths s consstent wth what Chrstensen and Greene found. The fact that we reject both the hpothess of homothetct and the hpothess of homogenet mples that a model that allows non-homothetct and non-homogenet n output s requred to represent the structure of productons for U.S. frms generatng electrc power. 7 Just lke n the prevous two estmatons, we are also nterested n the returns-to-scale. Here t s calculated as follows: r ln c ln ˆ β + * ˆ β ln + ˆ β ln pl + ˆ β Y LY Y ln pk Usng ths formula and plottng the values aganst the output gves a pattern depcted n Fgure. Agan, the graph tells us that n general, the electrc utlt frms exhbt an ncreasng returns-to-scale (r s greater than one), whle the economes-of-scale 8 falls wth output level. However, ths economes-of-scale s exhausted at output levels of around 3, mllon kwh (that s, at lnkwh 9.5 on the graph). Fgure. Returns-to-Scale and Output, C-G s Model A.89.5 rtsfull output Note: rtsfull: returns-to-scale for full model,.e. Model A. 7 Clearl, the Cobb-Douglas functon n Secton s a lmtng form of ths generalzaton n whch we mplctl mpose homothetct and homogenet. 8 Here, followng Chrstensen and Greene s defned as e /r. However, we should note that * ˆ β γ 9