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Working Paper 8702 ESTIMATING TOTAL FACTOR PRODUCTIVITY I N A GENERALIZED COST SYSTEM by Phi l i p I s r a i l e v i c h and K. J. Kowalewski P h i l i p I s r a i l e v i c h and K.J. Kowalewski are economists a t the Federal Reserve Bank o f Cleveland. Working papers o f t h e Federal Reserve Bank o f Cleveland a r e p r e l i m i n a r y m a t e r i a l s c i r c u l a t e d t o s t i m u l a t e discussion and c r i t i c a l comment. The views s t a t e d h e r e i n a r e those o f the authors and n o t n e c e s s a r i l y those o f the Federal Reserve Bank o f Cleveland o r o f t h e Board o f Governors o f the Federal Reserve System. March 1987 ESTIMATING TOTAL FACTOR PRODUCTIVITY IN A GENERALIZED COST SYSTEM I. Introduction Measuring the impact of regulation on electric utilities is of considerable interest to economists and regulators. Courville (1974), Spann (1 974), Peterson (1 975), Cowing (1 978), and Nel son and Nohar ( 1983>,for example, find evidence of an overcapitalization bias using variations of the Averch and Johnson (1962) model. These models are criticized by Atkinson and Halvorsen (1984) because the impact of additional regulatory constraints is ignored. Atkinson and Halvorsen (A-H) estimate a generalized cost model with a cross-section of electric utilities and find a regulatory bias on the utilization of noncapital factor inputs. Moreover, Fare and Logan (1983) show that a rate-of-return constraint alone invalidates the use of Shephard's Lemma for noncapital inputs. Joskow (1974) argues that regulation may also bias the rate of technical change implemented by utilities. The only empirical paper to have considered such an impact is Nelson and Wohar. They find a regulatory impact on the total factor productivity (TFP) experienced by electric utilities during the late 1970s. However, in addition to considering only'a rate-of-return constraint, their approach can be criticized because it considers the regulatory impact on TFP to be independent of the returns-to-scale and technical- change components o f TFP. ' Because TFP i s t h e sum o f r e t u r n s - t o - scale and technical- change terms i n a cost f u n c t i o n framework, Wohar model i m p l i e s a deus ex machina r e g u l a t o r y impact on TFP. t h e Nelson and Obviously, the authors cannot t e s t f o r a r e g u l a t o r y impact on t e c h n i c a l change. I n t h i s paper we develop a model t h a t can be used w i t h time- series d a t a t o t e s t f o r a r e g u l a t o r y impact on the components of TFP. The f o u n d a t i o n o f our model i s the A-H generalized c o s t f u n c t i o n , modified w i t h time v a r i a b l e s t o capture the dynamic aspect o f TFP. To t h i s , we add an equation f o r TFP. Although t h i s equation i s n o t necessary t o t e s t f o r t h e impact o f r e g u l a t i o n on TFP and i t s components, i t s use w i l l presumably increase t h e e f f i c i e n c y o f parameter estimates because i t i s a d d i t i o n a l behavioral i n f o r m a t i o n and because TFP i s measured as a r a t e o f change. Two a l t e r n a t i v e TFP equations are derived; one i s considerably e a s i e r t o estimate than the o t h e r . I n the n e x t s e c t i o n o f t h i s paper, an a l t e r n a t i v e d e r i v a t i o n o f the A-H model i s given. discussed. A f t e r t h a t , t h e two TFP equations a r e formulated and F i n a l l y , the t r a n s l o g version o f our generalized c o s t model i s given . 1. The TFP equation i n the Nelson and Wohar paper i s : where W = r a t e o f change o f TFP V T = t e c h n i c a l change v, = r e t u r n s t o scale Q = output M i = i - t h i n p u t share P i = input price s = rate o f return X = capital input C = cost. 11. The Atkinson and Halvorsen Model A t k i nson and Hal vorsen show t h a t general r e g u l a t o r y c o n s t r a i n t s a1 t e r t h e nature o f the cost- minimizing, f i r s t - o r d e r c o n d i t i o n s . I n s t e a d o f equating the marginal costs o f each f a c t o r i n p u t t o i t s market p r i c e , a r e g u l a t e d f i r m f i n d s i t optimal t o equate the marginal costs o f each f a c t o r i n p u t t o i t s shadow p r i c e . These shadow p r i c e s are market p r i c e s adjusted f o r the impact of r e g u l a t i o n , and t h e i r s p e c i f i c a t i o n depends on the exact nature of t h e regulatory constraints. p r i c e s , P:, Atkinson and Halvorsen approximate these shadow w i t h simple p r o p o r t i o n a l r e l a t i o n s h i p s t o market p r i c e s , P: = k i p i , f o r each i n p u t i. The generalized, o r shadow, c o s t f u n c t i o n i s simply the a c t u a l c o s t f u n c t i o n , b u t w i t h shadow p r i c e s i n s t e a d o f market Because shadow costs a r e n o t observed, t h e shadow-cost f u n c t i o n must prices. be r e w r i t t e n i n terms o f observable v a r i a b l e s . Accounting i d e n t i t i e s f o r t h e actual (Ca> and shadow (C') t o t a l costs are r e s p e c t i v e l y : (1) C a = C P i X I and where X i i s an i n p u t f a c t o r and P i i s i t s market p r i c e . The shadow-cost- share equations are: M: = kiPiX, ------C' f o r each i. Instead of the t r a d i t i o n a l system o f share equations, the f o l l o w i n g system can be derived: (2) P i x i = CSM7 -----, kl f o r each i . . . .- The sum o f these equations i s : and t a k i n g t h e l o g a r i t h m o f both sides y i e l d s : Equation ( 3 ' ) i s estimable a f t e r s u b s t i t u t i n g the shadow t r a n s l o g c o s t f u n c t i o n f o r lnC s and t h e d e r i v e d shadow c o s t shares MQ = a l n C s / a l n ( k l P I ) . Observable cost- share equations can be d e r i v e d by d i v i d i n g b o t h sides o f equation (2) by C a : where MB i s the a c t u a l c o s t share f o r i n p u t i, and by s u b s t i t u t i n g equation (3) i n t o equation ( 4 ) t o o b t a i n : (5) M? = kylMQ/Ckf 'M:. The A-H shadow-cost system i s ( 3 ' ) and (5). Because the sum o f t h e a c t u a l cost shares i s one, one o f the actual cost- share equations i n equation (5) can be dropped. 111. The Total Factor P r o d u c t i v i t y Equations The c o s t f u n c t i o n employed by Atkinson and Halvorsen uses o n l y shadow i n p u t f a c t o r p r i c e s and o u t p u t as arguments. I n order t o estimate TFP u s i n g the A-H model, time must be added as an a d d i t i o n a l explanatory v a r i a b l e . The d e r i v a t i v e o f shadow c o s t w i t h respect t o time i s then: The f i r s t term on the r i g h t - h a n d side can be s i m p l i f i e d by d i v i d i n g b o t h sides of the equation by C s , by u s i n g Shephard's Lemma f o r the shadow c o s t f u n c t i o n (i.e., .(aCs/aP:) = X,), and by n o t i n g t h a t k , i s constant. Then m u l t i p l y and d i v i d e the second term by Q t o obtain: where the s u p e r s c r i p t " i n d i c a t e s the r a t e o f change, f o r example 0 C s = (dlnCs/dT). respect t.0 output. v$ i s the shadow e l a s t i c i t y o f shadow c o s t w i t h v: i s u s u a l l y c a l l e d t e c h n i c a l change. I t i s the r a t e o f change o f shadow cost, h o l d i n g constant a l l cost f u n c t i o n arguments except time. can be defined s i m i l a r l y t o t h a t The r a t e o f change i n shadow TFP ( W S ) o f a c t u a l TFP using a D i v i s i a index o f factor- input- shadow shares Using t h e accounting i d e n t i t y ( 1 ) W s can be expressed as: Equation (7) i s analogous t o the equation derived by Nelson and Wohar f o r the r a t e o f change i n actual TFP (Ha): The important d i f f e r e n c e between equations ( 7 ) and ( 7 ' ) i s t h a t equation ( 7 ) was d e r i v e d by a p p l y i n g Shephard's Lemma t o shadow cost, w h i l e ( 7 ' ) uses Shephard's Lemma f o r actual c o s t . The difference between W a , defined i n ( 7 ' > , and W s can be d e r i v e d 0 analytically. S u b s t i t u t i n g the i d e n t i t y C a 0 = CS 0 + (Ca/CS> i n t o ( 7 ' ) . . . The advantage o f using ( 1 2 ' ) i s t h a t i t i s considerably e a s i e r t o Ws. estimate than (9). The disadvantage o f using ( 1 2 ' ) i s t h a t , i n case o f k i = 1 , i t does n o t use an a c t u a l vaTue o f W a as i n equation (7'). Instead, W a A o derived f r o m ( 1 2 ' 1 , w i t h k i = 1, i s W a = C a - IM;Pi, A where the M? are estimated actual cost shares. The Translog S p e c i f i c a t i o n o f t h e Generalized Cost System IV. The t r a n s l o g form o f the shadow-cost f u n c t i o n i s : (13) lnCS = a,, + C B l l n k i P l + BQlnQ+ RTT + .5CCyi , l n k i P i l n k J P J + C y l p l n k l P l l n Q + C y i ~ ( l n k . ~ P i )+T .5yQQ(lnQ)2+ yQT(lnQ)T + - ~ Y T T T ' . The .shadow-cost f u n c t i o n i s r e s t r i c t e d t o be 1 in e a r l y homogeneous w i t h r e s p e c t t o shadow p r i c e s using the c o e f f i c i e n t r e s t r i c t i o n s : (13') C O I = 1, CyiQ = 0, CyiT = 0, Cylj = 0, and y i j = y j i - The l o g a r i t h m i c p a r t i a l d e r i v a t i v e o f equation (13) w i t h respect t o l n k i P i , u s i n g the modified Shephard's Lemma, y i e l d s t h e t r a n s l o g cost- share equations: (14) M: = ( a l n C s ) / a l n ( k l P i ) S u b s t i t u t i n g equations (13) and (14) i n t o ( 3 ' ) y i e l d s t h e t r a n s l o g v e r s i o n o f the cost f u n c t i o n : The t r a n s l o g cost- share equations are o b t a i n e d by s u b s t i t u t i n g e q u a t i o n (14) i n t o (5): The r e t u r n s t o scale (v;), dC(Ml,/kl>/dT t e c h n i c a l change ( v ? ) , and expression for the t r a n s l o g shadow TFP equations (9) and ( 1 2 ' ) are: and Also, f o r t h e f o l l o w i n g discussion, note t h a t : . (18) v; = < a l n C a > / a l n Q= v; + (Ck;iyio)/Ck~'M:, and Thus, equation (9) can be r e w r i t t e n as: 0 (20) Ca 0 = vGQ + 0 V: + CMfP,. S i m i l a r l y , the t r a n s l o g form o f t h e second shadow TFP equation i s obtained by s u b s t i t u t i n g (17), (17'1, and (14) i n t o (12'1, which i s o b v i o u s l y a much s h o r t e r expression than equation (20) i n t r a n s l o g form. V. Estimating t h e Regulatory Bias Atkinson and Halvorsen showed t h a t equations ( 1 5 5 and (16) are homogeneous o f degree zero (h.d.2.) w i t h r e s p e c t t o the k i . Therefore, one o f k , can be chosen a r b i t r a r i l y , and one i s a n a t u r a l and convenient n o r m a l i z a t i o n value. An estimate o f the e f f e c t o f r e g u l a t i o n on t o t a l c o s t and o t h e r components i s obtained by comparing the f i t t e d values o f the d e s i r e d v a r i a b l e generated by the estimated model, w i t h a1 1 of the k , equal t o t h e i r estimated values (estimated r e g u l a t o r y impact included), w i t h the f i t t e d values o f the same v a r i a b l e b u t w i t h a l l o f the k , s e t t o one (no regulation). I t i s important t o note t h a t t h i s procedure works o n l y f o r v a r i a b l e s whose equations are h.d.z. w i t h r e s p e c t t o t h e k , ; otherwise, the magnitude o f t h e r e g u l a t o r y b i a s depends on t h e value o f the k , n o r m a l i z a t i o n . The lnC S equation (13) i s n o t h.d.z. i n the k , . associated w i t h t h e y , c o e f f i c i e n t s a r e h.d.z. A l l o f the terms w i t h respect t o k , from t h e c o e f f i c i e n t r e s t r i c t i o n s (13'1, b u t the terms r e l a t e d t o the 13, c o e f f i c i e n t s a r e not; i f the k i are m u l t i p l i e d by some constant t, then I B l l n t = l n t using (13'). The shadow share equations (14) are h.d.2. no terms i n v o l v i n g b o t h the B h.d.2. and k i . Hence, t h e equations (16) a r e w i t h respect t o the k i because the t f a c t o r s f o r the kT1 w i l l cancel o u t i n t h e numerator and denominator. h.d.2. i n the k l because they have Actual c o s t (equation (15)) is.-., i n the k i because the l n t term f o r t h e nonhomogeneous component of lnCs w i 11 cancel o u t w i t h the -1nt term o f the nonhomogeneous component of lnC(Mi,/kl>. Both v$ and v: are h.d.2. terms. i n v o l v i n g both the B1 and k i ; v: are h.d.2. i n the k l because they have no and vS (equations (18) and (19)) i n the k i because t h e terms added t o v$ and v9 are h.d.2. in the k l . Both TFP equations, (20) and ( 1 2 ' ) , a r e h.d.2. components o f (20) are h.d.2. k i because M: i s h.d.2. i n the k i . The f i r s t two i n k i ; the t h i r d term a l s o i s h.d.2. i n the k i . have already been shown t o be h.d.z. i n the A l l o f the terms i n equation (10) i n the k i . Thus, the e f f e c t of r e g u l a t i o n on TFP i s computed by adding the i n d i v i d u a l r e g u l a t o r y e f f e c t s on r e t u r n s t o scale, t e c h n i c a l change, and shadow shares. This i s an improvement over the Nelson and Wohar approach, which assumed t h a t returns- to- scale and technical- change components a r e independent o f any r e g u l a t o r y e f f e c t . References A t k i n s o n , S c o t t E., and R o b e r t Halvorsen. " Parametric E f f i c i e n c y Tests, Economies o f Scale, and I n p u t Demand i n U.S. E l e c t r i c Power Generation," I n t e r n a t i o n a l Economic Review, v o l . 25, no. 3 (October 19841, 647-62. Averch, Harvey, and Leland L. Johnson. " Behavior o f t h e F i r m under R e g u l a t o r y C o n s t r a i n t , " American Economic Review, v o l . 52, no. 5 (December 19621, 1052-69. C o u r v i l l e , Leon. " R e g u l a t i o n and E f f i c i e n c y i n t h e E l e c t r i c U t i l i t y I n d u s t r y , " The B e l l J o u r n a l of Economics and Management Science, v o l . 5, no. 1 ( S p r i n g 19741, 53-74. Cowing, Thomas G. "The E f f e c t i v e n e s s o f Rate- of- Return Regulation: An E m p i r i c a l T e s t Using P r o f i t Functions," i n M. Fuss and D. McFadden, Eds., p r o d u c t i o n Economi cs : A Dual Approach t o Theory and Appl ic a t i o n , Amsterdam: .North H o l l a n d P u b l i s h i n g Company, 1978. Fare, R o l f , and James Logan. "Shephard's Lemma and Rate o f Return R e g u l a t i o n , " Economics L e t t e r s , v o l . 12 (19831, 121-25. Joskow. Paul L. " I n f l a t i o n and Environmental Concern: S t r u c t u r a l Chanae i n t h e Process o f Pub1 i c U t i 1 it y P r i c e R e g u l a t i o n , " The J o u r n a l o f L& and Economics, v o l . 17 (October 19741, 291-327. Nelson, Randy A., and Mark E. Wohar. " R e g u l a t i o n , Scale Economies, and P r o d u c t i v i t y i n Steam- Electric Generation," I n t e r n a t i o n a l Economic Review, v o l . 24, no. 1 (February 19831, 57-79. Petersen, H . C r a i g . "An E m p i r i c a l Test o f R e g u l a t o r y E f f e c t s , " The B e l l Journal o f Economics and Management Science, v o l . 6, no. 1 ( S p r i n g 19751, 11 1-26. Spann, R.M. "Rate o f Return R e g u l a t i o n and E f f i c i e n c y i n P r o d u c t i o n : An E m p i r i c a l T e s t of t h e Averch- Johnson Thesis," The B e l l Journal o f Economics and Management Science, v o l . 5, no. 1 ( S p r i n g 19741, 38-52.
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