A description of two ways of adding an equation for total factor productivity to the generalized cost system of Atkinson and Halvorsen; includes a discussion of testing for regulatory biases.Electric utilities ; Industrial policy

K.J. Kowalewski

Philip Israilevich

English

Research Papers in Economics

Working Paper  8702 
ESTIMATING  TOTAL  FACTOR  PRODUCTIVITY 
IN A  GENERALIZED  COST  SYSTEM 
by  Phi lip Israi  levich and  K. J.  Kowalewski 
Philip Israilevich and  K.J.  Kowalewski  are 
economists  at the Federal  Reserve  Bank  of 
Cleveland. 
Working  papers  of the Federal  Reserve  Bank  of 
Cleveland are preliminary materials 
circulated to  stimulate discussion  and 
critical comment.  The  views  stated herein 
are those  of the authors and  not necessarily 
those of the Federal  Reserve  Bank  of 
Cleveland or of the Board  of Governors  of 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  of TFP. '  Because  TFP  is the sum  of  returns-to- 
scale and  technical-change terms  in  a cost function framework,  the Nelson  and 
Wohar  model  implies a deus  ex  machina regulatory impact on  TFP.  Obviously, 
the authors cannot  test for a regulatory impact on  technical  change. 
In  this paper  we  develop a model  that can  be  used  with time-series data to 
test for a regulatory impact on  the components  of  TFP.  The  foundation of 
our model  is  the A-H  generalized cost function, modified  with time  variables 
to capture  the dynamic  aspect  of TFP.  To  this,  we  add  an  equation for TFP. 
Although  this equation is  not necessary  to test for the impact  of  regulation 
on  TFP  and  its  components,  its use  will presumably  increase  the efficiency of 
parameter  estimates because it  is  additional  behavioral  information and 
because  TFP  is  measured  as  a rate of change.  Two  alternative TFP  equations 
are derived;  one  is  considerably easier to estimate  than the other. 
In  the next section of this paper,  an  alternative derivation of the  A-H 
model  is given.  After that,  the two TFP  equations  are formulated and 
discussed.  Finally,  the  translog version of our  generalized cost model  is 
given  . 
1.  The  TFP  equation  in the Nelson  and  Wohar  paper  is: 
where  W  =  rate of change  of TFP 
VT =  technical  change 
v,  =  returns  to scale 
Q  =  output 
Mi =  i-th input share 
Pi =  input price 
s  =  rate of return  - 
X  =  capital  input 
C  =  cost. 11.  The  Atkinson and  Halvorsen Model 
Atki  nson  and  Hal vorsen  show  that general  regulatory constraints a1  ter the 
nature of the cost-minimizing,  first-order conditions.  Instead of equating 
the marginal  costs of  each  factor input to its  market  price,  a regulated firm 
finds it  optimal  to  equate  the marginal  costs of each  factor input to its 
.  . .- 
shadow  price.  These  shadow  prices are market  prices adjusted for the  impact 
of  regulation,  and  their specification depends  on  the exact nature of  the 
regulatory constraints.  Atkinson and  Halvorsen approximate  these  shadow 
prices,  P:,  with simple proportional relationships  to  market  prices, 
P:  =  kipi, for each  input i. The  generalized,  or shadow,  cost function is 
simply  the actual cost function,  but with shadow  prices instead of  market 
prices.  Because  shadow  costs  are not observed,  the shadow-cost  function must 
be  rewritten in  terms  of observable variables. 
Accounting identities for the actual  (Ca> and  shadow  (C')  total costs 
are respectively: 
(1)  C
a  =  CPiXI  and 
where  Xi  is  an  input factor and  Pi  is  its  market  price.  The  shadow-cost- 
share  equations  are: 
kiPiX, 
M:  = -------  for each i. 
C ' 
Instead of  the  traditional  system of share  equations,  the following system  can 
be  derived: 
CSM7 
(2)  Pixi = -----,  for each i. 
k  l The  sum  of these  equations  is: 
and  taking the logarithm of  both sides  yields: 
Equation  (3') is estimable  after  substituting the  shadow  translog cost 
function for lnC
s  and  the derived shadow  cost shares  MQ  =  alnCs/aln(klPI). 
Observable  cost-share equations  can  be  derived by  dividing both sides of 
equation  (2) by  C
a: 
where  MB  is  the actual  cost share  for input i,  and  by  substituting 
equation  (3)  into equation  (4) to  obtain: 
(5)  M?  =  kylMQ/Ckf  'M:. 
The  A-H  shadow-cost  system  is  (3') and  (5).  Because  the  sum  of the actual 
cost  shares  is  one,  one  of the actual  cost-share equations  in  equation  (5)  can 
be  dropped. 
111.  The  Total  Factor  Productivity Equations 
The  cost function employed  by  Atkinson and  Halvorsen  uses  only shadow 
input factor prices and  output as  arguments.  In  order  to estimate  TFP  using 
the  A-H  model,  time must  be  added  as  an  additional  explanatory variable.  The 
derivative of shadow  cost with respect to time  is  then: 
The  first term on  the right-hand side can  be  simplified by  dividing both sides 
of  the  equation by  C
s,  by  using Shephard's  Lemma  for the  shadow  cost function (i  .e.,  .(aCs/aP:)  =  X,),  and  by noting that k,  is constant. 
Then  multiply and  divide the  second  term by Q  to  obtain: 
where  the superscript "  indicates the  rate of change,  for example 
0 
C
s  =  (dlnCs/dT).  v$ is  the  shadow  elasticity of shadow  cost with  .  . . 
respect t.0 output.  v:  is  usually called technical  change.  It  is  the 
rate of change  of shadow  cost,  holding constant all cost function arguments 
except  time. 
The  rate of change  in shadow  TFP  (WS)  can  be  defined similarly to that 
of actual TFP  using a Divisia index of factor-input-shadow shares 
Using the accounting identity (1) 
W
s can  be  expressed  as: 
Equation  (7)  is  analogous  to the equation derived by  Nelson  and  Wohar  for 
the rate of change  in  actual  TFP  (Ha): 
The  important  difference between  equations  (7) and  (7') is  that equation  (7) 
was  derived by  applying Shephard's  Lemma  to shadow  cost,  while  (7') uses 
Shephard's  Lemma  for actual  cost. 
The  difference  between  W
a,  defined  in (7'>, and  W
s can  be  derived 
0  0  0 
analytically.  Substituting the  identity C
a  =  C
S  +  (Ca/CS> into (7') Ws.  The  advantage of using (12')  is that it  is  considerably easier to 
estimate  than  (9).  The  disadvantage of  using (12')  is that,  in  case  of ki =  1, 
it  does  not use  an  actual  vaTue  of  W
a as  in  equation  (7').  Instead,  W
a 
o  A  A 
derived from (12'1,  with ki =  1,  is  W
a =  C
a - IM;Pi,  where  the  M? 
are estimated actual  cost  shares. 
IV.  The  Translog Specification of the Generalized Cost  System 
The  translog form of the  shadow-cost  function is: 
(13)  lnCS =  a,,  +  CBllnkiPl +  BQlnQ  +  RTT 
+  .5CCyi ,lnkiPilnkJPJ +  CylplnklPllnQ 
+  Cyi~(lnk.~Pi)T  +  .5yQQ(lnQ)2  +  yQT(lnQ)T +  -~YTTT'. 
The .shadow-cost  function i  s restricted to be  1 i  nearly homogeneous  wi  th respect 
to shadow  prices using the coefficient restrictions: 
(13')  COI  =  1,  CyiQ =  0,  CyiT  =  0, 
Cylj =  0,  and  yij =  yji- 
The  logarithmic partial derivative of equation  (13)  with respect  to 
lnkiPi, using the modified  Shephard's  Lemma,  yields the  translog 
cost-share  equations: 
(14)  M:  =  (alnCs)/aln(klPi) 
Substituting equations  (13)  and  (14)  into (3')  yields  the  translog version of 
the  cost function: The  translog cost-share equations are obtained by  substituting equation  (14)  . 
into (5): 
The  returns to scale  (v;),  technical  change  (v?),  and 
dC(Ml,/kl>/dT  expression for the  translog shadow  TFP equations  (9)  and 
(12')  are: 
and 
Also,  for the following discussion,  note  that: (18)  v;  =  <alnCa>/alnQ  =  v;  +  (Ck;iyio)/Ck~'M:,  and 
Thus,  equation  (9)  can  be  rewritten as: 
0  0  0 
(20) C
a  =  vGQ  +  V:  +  CMfP,. 
Similarly,  the  translog form of the second  shadow  TFP  equation  is  obtained 
by  substituting (17),  (17'1,  and  (14)  into (12'1,  which  is  obviously a much 
shorter expression than equation  (20)  in  translog form. 
V.  Estimating the Regulatory Bias 
Atkinson and  Halvorsen  showed  that equations  (155  and  (16)  are homogeneous 
of  degree  zero  (h.d.2.)  with respect to the  ki.  Therefore,  one  of k, can 
be  chosen  arbitrarily, and  one  is a natural  and  convenient  normalization 
value.  An  estimate of the effect of regulation on  total cost and  other 
components  is  obtained by  comparing  the fitted values  of the desired variable 
generated  by  the estimated model,  with a1 1 of  the k, equal  to their 
estimated values  (estimated regulatory impact  included),  with the fitted 
values of the  same  variable but with all of the k, set to  one  (no 
regulation). 
It is  important  to  note  that this procedure works  only for variables whose 
equations  are  h.d.z.  with respect to the k,;  otherwise,  the magnitude  of the 
regulatory bias depends  on  the value of the  k, normalization. 
The  lnC
S  equation  (13)  is not h.d.z.  in  the k,.  All of the  terms 
associated with the y, coefficients are h.d.z.  with respect  to  k, from 
the coefficient restrictions (13'1,  but the  terms  related to the  13, coefficients are not;  if the  ki are multiplied by  some  constant t, then 
IBllnt  =  lnt  using (13'). 
The  shadow  share  equations  (14)  are h.d.2.  in  the  kl because  they have 
no terms  involving both the  B  and  ki  .  Hence,  the equations  (16)  are 
h.d.2.  with respect  to  the ki because  the t factors for the kT1 will 
cancel  out in  the numerator  and  denominator.  Actual  cost (equation  (15))  is.-., 
h.d.2.  in  the ki because  the  lnt term for the nonhomogeneous  component  of 
lnCs wi  11  cancel  out with the -1nt  term of the nonhomogeneous  component  of 
lnC(Mi,/kl>.  Both  v$  and  v:  are  h.d.2.  in  the kl because  they  have  no 
terms. involving both the B1 and  ki; v:  and  vS  (equations  (18)  and  (19)) 
are h.d.2.  in  the  ki because  the terms  added  to v$ and  v9  are  h.d.2.  in 
the kl. 
Both  TFP  equations,  (20)  and  (12'),  are h.d.2.  in the ki.  The  first two 
components  of (20)  are h.d.2.  in  ki; the third term also is  h.d.2.  in  the 
ki because  M:  is h.d.2.  in  the ki.  All of the  terms  in  equation  (10) 
have  already been  shown  to be  h.d.z.  in the ki.  Thus,  the effect of 
regulation on  TFP  is  computed  by  adding  the  individual  regulatory effects on 
returns to scale,  technical  change,  and  shadow  shares.  This  is  an  improvement 
over the Nelson  and  Wohar  approach,  which  assumed  that returns-to-scale and 
technical-change components  are independent  of any  regulatory effect. References 
Atkinson,  Scott E.,  and  Robert Halvorsen.  "Parametric  Efficiency Tests, 
Economies  of  Scale,  and  Input Demand  in  U.S.  Electric Power  Generation," 
International  Economic  Review,  vol.  25,  no.  3  (October  19841,  647-62. 
Averch,  Harvey,  and  Leland L.  Johnson.  "Behavior of the Firm under  Regulatory 
Constraint,"  American  Economic  Review,  vol.  52,  no.  5  (December  19621, 
1052-69. 
Courville,  Leon.  "Regulation and  Efficiency in  the Electric Utility Industry," 
The  Bell Journal  of  Economics  and  Management  Science,  vol.  5,  no.  1 
(Spring 19741,  53-74. 
Cowing,  Thomas  G.  "The  Effectiveness of Rate-of-Return Regulation:  An 
Empirical  Test Using Profit Functions,"  in  M.  Fuss  and  D.  McFadden,  Eds., 
product  ion Economi cs : A  Dual  Approach  to Theory  and  Appl i  cation, 
Amsterdam:  .North Holland Publishing Company,  1978. 
Fare,  Rolf,  and  James  Logan.  "Shephard's  Lemma  and  Rate  of Return Regulation," 
Economics  Letters,  vol.  12  (19831,  121-25. 
Joskow.  Paul  L.  "Inflation and  Environmental  Concern:  Structural Chanae  in 
the Process  of Pub1  ic  Uti  1 i  ty  Price Regulation,"  The  Journal of L&  and 
Economics,  vol.  17  (October  19741,  291-327. 
Nelson,  Randy  A.,  and Mark  E.  Wohar.  "Regulation,  Scale  Economies,  and 
Productivity in  Steam-Electric Generation,"  International  Economic  Review, 
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Petersen,  H.  Craig.  "An  Empirical  Test of  Regulatory Effects,"  The  Bell 
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Spann,  R.M.  "Rate of  Return Regulation and  Efficiency in  Production:  An 
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Behavior of the Firm under Regulatory Constraint,&quot;

The Effectiveness of Rate-of-Return Regulation: An Empirical Test Using Profit Functions,&quot;

Estimating total factor productivity in a generalized cost system

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Estimating total factor productivity in a generalized cost system

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