Raa, M.H. ten

English

Research Papers in Economics

JOURNAL  OF  ECONOMIC  THEORY  31,  176-181  (1983) 
Supportability  and  Anonymous  Equity* 
THIJS  TEN  RAA 
New  York  University,  New  York,  New  York,  10003,  and 
Erasmus  University,  3OOODR  Rotterdam,  The  Netherlands 
Received  June  26,  1981:  revised  October  14,  198  I 
W.  W.  Sharkey  and  L.  G.  Telser  (J.  Econom.  Theory  18  (1978).  23-37)  feel  that 
invulnerability  of  a  natural  monopoly  to  the  threat  of  competitive  entry  is  well 
reflected  in  the  concept  of  supportability.  G.  R.  Faulhaber  and  S.  B.  Levinson 
(Amer.  Econom.  Rev.  71  (1981),  1083-1091)  point  out  that  supportability  is 
necessary  for  the  achievability  of  anonymous  equity,  i.e.,  absence  of  consumer 
subsidies  in  public  enterprise  pricing.  This  paper  reconciles  supportability  with 
market  clearance  and  shows  that  supportability  is  sufficient  for  the  achievability  of 
anonymous  equity.  Journal  of  Economic  Literature  Classification  Numbers:  022. 
611,  614. 
A  cost function  is supportable at  an output  vector  x0  if prices exist which 
cover  total  costs and  render  supply  of  any  part  of  the  output  x  <x0 
unprofitable.  (Precise definitions  will  be  presented in  the  next  section.)  A 
cost  function  is supportable if  it  is supportable  at  any  output  vector.  The 
supporting  prices clearly  have  to  do  with  invulnerability  to  the  threat  of 
entry.  But  do  they  call  forth  the  output  under  consideration:  x0;  are 
supporting prices market  clearing? Sharkey  and Telser kill  this complication 
by  assuming  that  demand is completely  inelastic.  This paper will  relax  the 
complete inelasticity  assumption drastically.  If  there are threshold quantities 
of  demand for  all goods, then  supportability  and market  clearance can be 
reconciled. 
The  problem  also  arises in  another  related  context;  Willig  [6]  and 
Faulhaber  and  Levinson  ]2]  define  the  absence of  consumers subsidies  in 
public  enterprise  pricing  as anonymous  equity.  Prices  are  anonymously 
equitable  if  they  cover  total  costs of  the  quantities called forth  by  demand 
*  I  would  like  to  thank  Gerald  R.  Faulhaber  and  Jess  Benhabib  for  important  suggestions. 
The  paper  was  presented  at  the  European  Meeting  of  the  Econometric  Society,  Amsterdam, 
The  Netherlands,  1981.  Sloan  Foundation  support  is  gratefully  acknowledged. 
176 
0022-053  l/83  $3.00 
Copyright  $’  1983  by  Academic  Press.  Inc. 
All  rights  of  reproduction  in  any  form  reverved. SUPPORTABILITY  AND  ANONYMOUS  EQUITY  177 
and  any  part  of  demand  generates  revenues  no  greater  than  its  stand-alone 
costs.  Faulhaber  and  Levinson  observe  that  supportability  of  the  cost 
function  at  the  quantities  demanded  is  a necessary  condition  for  anonymous 
equity  to  be  achievable.  In  fact,  the  preceding  definitions  imply  that 
anonymous  equity  is  market  clearing  supportability  at  the  quantities 
demanded.  Hence  the  proclaimed  result  is  equivalent  to  the  following.  If 
there  are  threshold  quantities  of  demand  for  all goods,  then  supportability  is 
a sufficient  condition  for  anonymous  equity  to  be  achievable. 
In  the  recent  theories  of  natural  monopoly  and  public  enterprise  pricing 
two  elements  are  of  importance.  One  is  invulnerability  to  the  threat  of  entry 
or,  depending  on  the  setting,  lack  of  subsidization.  The  other  is  market 
clearance.  These  elements  seem  quite  different  if  not  independent.  The  result 
of  this  paper  indicates  though  that  the  second  element,  market  clearance,  is 
hidden  in  the  first  one,  invulnerability  or  subsidy-freeness.  Solution  concepts 
which  include  market  clearance,  such  as  anonymous  equity,  are  only  super- 
ficially  tighter  than  those  which  do  not,  such  as  supportability. 
PRELIMINARIES 
First  some  notation  and  definitions  are  copied  from  Sharkey  and  Telser 
ISI. R:  will  represent  the  nonnegative  orthant  of  n-dimensional  Euclidean 
space.  For  any  two  members  x  and y  the  expression  x  >  y  is  to  be interpreted 
xi > yi  for  i =  l,...,  n  and  x  > y  will  be  written  if  x  >  y  and  x  #  y. 
Furthermore,  the  expression  x  % y  is  to  be  interpreted  xi  > yi  for  i =  l,...,  n. 
The  inner  product  of  x  and  y  will  be  written  (x,  y)  =  Cr=,  xi yi.  Any 
nondecreasing  function  on  iR:  will  be  considered  a  cost  function.  A  cost 
function  c  is  subhomogeneous  if  c(Ax)  <  Lc(x)  for  all x  >  0 and  A>  1. A  cost 
function  c  is  supportable  at  x0  if  p(x”)  =  (p  E  iR:  1 (p,  x0)  =  c(x”)  while 
0 <x  <x0  implies  (p,x)  <  c(x)}  is  nonempty.  c  is  supportable  if  it  is 
supportable  at  any  x0  >  0.  The  latter  property  will  be briefly  referred  to  by 
the  notion  of  supportability.  Observing  axiomatic  value  theory,  a  demand 
correspondence  q  will  be  an upper  hemicontinuous  convex-valued  correspon- 
dence  from  iR:  to  IR:  .  (Upper  hemicontinuity  means  that  xm -+ x, 
y”  E  q(xm)  implies  ymk +  y,  y  E  q(x)  for  some  subsequence.  Upper  semicon- 
tinuity  means  that  xm -+x,  y”  E  4(x”),  y”  -+ y  implies  y  E  q(x).  Note  that 
upper  hemicontinuity  is  sufficient  for  upper  semicontinuity.)  By  definition, 
threshold  quantities  E for  all goods  fulfill  0 <  E <  q.  Following  Faulhaber  and 
Levinson  [ 21, anonymous  equity  is  said  to  be achievable  if there  are p”  E  R 1 
and  x0 E  IR$  with  p”  Ep(x’)  and  x0  E  q(p’).  In  other  words,  anonymous 
equity  is  achievable  if  and  only  if  there  is  market  clearance  and  suppor- 
tability  at  the  quantities  demanded. 
If  q  is  single-valued,  then  the  definition‘of  anonymous  equity  becomes. 178  THIJS  TEN  RAA 
substituting  the  second  condition  into  the  first  one:  there  is  a Q”  E  iR:  with 
(p”,  q(p’))  =  c(q(p’))  while  0 <x  Q q(p”)  implies  (p”,  x)  <  c(x).  In  this 
case  the  closely  related  concept  of  sustainability  is  defined  as  follows:  there 
is  a  p”  E  IR:  with  (p”,  q(p’))  =  c(q(pO)),  while  0 <p  ,<p”,  0 <x  <  q(p) 
implies  (p,  x)  <  c(x)  ( see  [2]  and  references  therein).  Reference  12, 
Proposition  71 neatly  organizes  the  solution  concepts  we  have  come  across. 
Supportability  at the  quantities  demanded  is  necessary  for  anonymous  equity 
to  be  achievable  which  on  its  turn  is  necessary  for  sustainability.  In  fact, 
from  supportability  at  the  quantities  demanded  one  obtains  anonymous 
equity  by  the  additional  requirement  of  market  clearance.  And  from 
anonymous  equity  one  obtains  sustainability  by  requiring  that  the  condition 
of  subsidy  freeness  or  invulnerability  to  the  threat  of  entry  is  also  met  for 
lower  prices.  The  first  tightening  up  is  but  superficial,  as  the  result  of  this 
paper  indicates.  The  second  restriction,  in  going  from  anonymous  equity  to 
sustainability,  is  an  open  issue.  This,  however,  is  not  so  much  a  matter  of 
market  clearance  and  beyond  the  scope  of  the  paper. 
ANALYSIS 
Our  strategy  of  proving  that  under  supportability  anonymous  equity  is 
achievable  consists  of  finding  a fixed  point  of  (p,  x)  tip(x)  x  q(p).  For  this 
purpose  we  want  the  constituent  mappings  to  be  upper  semicontinuous 
convex-valued  correspondences.  q  is  like  that  by  definition  but  for  p(a)  we 
have  to  prove  that  it  is  in  fact  an  upper  semicontinuous  convex-valued 
correspondence.  The  assumed  supportability  guarantees  the  existence  of p(a) 
as  a  correspondence.  Further,  it  is  convex-valued  by  definition.  Thus  the 
point  is  to  prove  upper  semicontinuity.  The  difficulty  is  that  only  suppor- 
tability  is  assumed  whereas  we  need  some  kind  of  regularity.  Lemma  1 will 
close  the  gap  by  proving  that  for  cost  functions  an  aspect  of  supportability 
(namely,  subhomogeneity)  implies  continuity.  Then  Lemma  2 will  finish  off 
by  showing  that  p(-)  is  as  desired.  In  this  way,  invoking  some 
straightforward  compactness  considerations  and  Kakutani’s  fixed  point 
theorem,  we  shall  find  a fixed  point  of  the  described  mapping  which  clearly 
proves  that  anonymous  equity  is  achievable  given  supportability. 
LEMMA  1.  A  subhomogeneous  cost  function  is  continuous  from  below.  It 
is  continuous  on  its  domain’s  interior. 
Proof:  Let  the  cost  function  be c  and  let  “1  3  xm +  x.  Define  1,  &,,)  as 
the  maximum  (minimum)  of  the  existing  xi/x:  and  unity.  Then  I,  1 1 and, 
for  x  9  0,  ,u,,, T 1.  For  large  m,  n,,,xT  >  xi >,u,xy.  This  follows  from  k,,,'s 
@,,,‘s)  maximizing  (minimizing)  property  for  xi  >  0 (XT  >  0)  and  is  otherwise SUPPORTABILITY  AND  ANONYMOUS  EQUITY  179 
obvious.  Thus  for  large  m,  A,,,xm > x  >pmxm.  It  follows  that  by 
nondecreasingness of  cost  function  c  and  subhomogeneity,  A;  ‘c(x)  < 
A;  l~(A,~m)  < c(xm) < ILI;‘c@,x~)  <PU, ‘c(x).  It  follows  that  c(x)  < 
lim c(xm) <  c(x)  where the second  inequality  is true for x  4  0.  Q.E.D. 
LEMMA  2.  For  a  supportable cost function,  p(a)  is an  upper semicon- 
tinuous convex-valued  correspondence.  It  is bounded on  (x0 E  IR:  1  x0 >  F) 
for  e 9 0. 
Proof  By  supportability,  p(.)  is  a  correspondence.  The  convex 
valuedness  is obvious. To  prove  upper semicontinuity,  let IR:  3 +yrn  -+ x0  > 0 
and p(x”)  3 pm  -+ p”.  Then 
POE  R’:.  (1) 
By  [S,  Proposition  21, the cost function,  say c,  is subhomogeneous,  and, 
by  Lemma 1, c  is continuous from  below.  Consequently, lim c(xm) >  c(x’), 
and, using pm  E p(x”), 
(PO5  x0> -  c(xO) >  lim [ ( pm, x”)  -  c(x*)]  >  0.  (2) 
For  0 ,< x  ,< x0  there  are y”  < xm  with  y”  T x.  Using pm up  and  the 
nondecreasingness  of  cost  function  c,  (p”,  y”)  <  c(y”)  <  c(x).  It  follows 
that 
I 
(PO1  x> <  c(x).  (3) 
By  (l),  (2),  and  (3),  p”  Ep(x’)  which  proves  the  upper  semicontinuity. 
Finally,  on  x0 >  E >>  0, p”  E p(x’)  implies (p”,  E) <  C(E) which  implies that 
p”  is bounded independently of x0  and therefore p(.)  is bounded.  Q.E.D. 
Now  we present our  main result. 
PROPOSITION.  If  there are threshold quantities of  demand  for  all goods, 
then  supportability  is  a  suflcient  condition  for  anonymous equity  to  be 
achievable. 
Proof.  By  Lemma  2,  a  convex  compacturn  9  contains 
p((x  E  IR;  ( x  > E}).  By  the upper hemicontinuity  of q,  a convex  compacturn 
d  contains q(Y).  By  the demand assumption d  can be situated in  (x  E iR:  1 
x>E}.  Now  take  p E 9  and  XE  6.  It  follows  that  p(x)cp(d)c 
p({x  E  IR:  ] x  >  E)) c  9  and q(p)  c  q(Y)  c  6.  Consequently, by  Lemma 2 
and  Kakutani’s  fixed  point  theorem,  9  x  d  3  (p, x)  bp(x)  x  q(p)  c 
.Y  xd  has  a  fixed  point  (p”,  x0)  with  p”  Ep(x’)  c  IR:  and 
x0 E q(p0) C w;.  Q.E.D. 180  THIJS  TEN  RAA 
Our  demand assumption merely  rules out  a boundary  complication  which 
is independent and has been analyzed  in  [ 11. An  alternative  set of  demand 
assumptions  which  yields  the  same  result  consists  of  weak  gross 
substitutability  and normality  as defined in  [4]  along with  the condition  that 
every  output  ray  (consisting of  bundles with  fixed  proportions)  contains a 
profitable  point.  This  can  be proved  in  the  same way  as [3,  Theorem 3 ]. 
Both  the  threshold  and  the  profitability  assumptions prevent  the  solution 
from  degenerating into  the  trivial  one that  reflects unwillingness to  pay  the 
costs of  the goods. 
DISCUSSION 
An  anonymous  referee made  an  interesting  comment.  (S)he  interpreted 
that  if  the cost function  is supportable then any  output  is stable in  a certain 
sense  provided  that  demand is completely  inelastic, but  otherwise there is at 
least one stable output  by  our proposition.  But  (s)he wondered if  there is any 
way  to  identify  the  anonymously  equitable  prices and  quantities.  Here  I 
would  like  to  make two  remarks. One  is on the nature  of  the  problem. The 
issue  is  to  tind,  simultaneously,  p”  Ep(x’)  and  x0 E q(p’).  These 
relationships are,  respectively,  a  supply  schedule and  a demand  schedule. 
The  problem  is essentially  to  find  the  intersection  of  these schedules. The 
appropriate  tools  are the  approximation  techniques of  equilibrium  analysis. 
The other  remark  is a brief  reference to  [2,  Proposition  91. This proposition 
provides  conditions  under  which  the  Ramsey  optimum  is  anonymously 
equitable. 
CONCLUSION 
The entry  concept of  supportability  and the equilibrium concept of market 
clearance can be reconciled. For  anonymous equity  to be achievable, suppor- 
tability  is not  only  necessary as pointed out  by  Faulhaber  and Levinson  [2] 
but  also sufficient  provided  that  there are threshold quantities of demand for 
all goods. 
REFERENCES 
I.  G.  DEBREU,  New  concepts  and  techniques  for  equilibrium  analysis,  Infernat.  Econom.  Rev. 
3  (1962).  257-273. 
2.  G.  R.  FAULHABER  AND  S.  B.  LEVINSON,  Subsidy-free  prices  and  anonymous  equity,  Amer. 
Econom.  Rev.  71  (1981).  1083-1091. SUPPORTABILITY  AND  ANONYMOUS  EQUITY  181 
3.  TH.  TEN  RAA,  “A  Theory  of  Value  and  Industry  Structure,”  Ph.D.  dissertation,  New  York 
University,  198  1. 
4.  I.  W.  SANDBERG,  Two  theorems  on  a  justification  of  the  multiservice  regulated  company, 
Bell  J.  Econom.  6  (1975).  346-356. 
5.  W.  W.  SHARKEY  AND  L.  G.  TELSER,  Supportable  cost  functions  for  the  multiproduct  firm, 
J.  Econom.  Theory  18  (1978)  23-37. 
6.  R.  D.  WILLIG.  Customer  equity  and  local  measured  service,  in  “Perspectives  on  Local 
Measured  Service”  (J.  A.  Baude  et  al.,  Eds.),  Kansas  City,  Kansas.  1979. 

Supportability and anonymous equity.

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