Muysken, Joan

English

Research Papers in Economics

Economics  Letters  11  (1983)  217-221 
North-Holland  Publishing  Company 
217 
TRANSFORMED  BETA-CAPACITY  DISTRIBUTIONS  OF 
PRODUCTION  UNITS  * 
Joan  MUYSKEN 
University  of  Groningen,  9700  A V  Groningen,  The  Netherlands 
Received  29  April  1982 
In  this  paper  the  capacity  distributions  corresponding  to  several  well-known  aggregate 
production  functions  (Cobb-Douglas,  CES  and  VES)  are  derived.  It  is  shown  that  these 
distributions  can  be  transformed  to  beta-distributions  in  a  straightforward  way. 
1. The  distribution  approach 
According  to  the  distribution  approach  to  the  aggregation  of  produc- 
tion  functions,  each  different  form  of  the  aggregate  short-run  production 
function  of  an  industry  corresponds  to  a  different  form  of  the  capacity 
distribution  of  production  units  in  that  industry  [Johansen  (1972),  Sato 
(1975)].  In  this  paper  we  shall  derive  the  capacity  distributions  corre- 
sponding  to  several  well-known  aggregate  production  functions  and 
investigate  their  properties.  It  will  be  shown  that  all  derived  capacity 
distributions  can  be  transformed  to  beta-distributions  in  a  straightfor- 
ward  way.  The  beta-distribution  therefore  appears  to  be  a  rather  general 
form  for  the  capacity  distribution.  This  is  also  interesting  since  the 
beta-distribution  allows  for  a  bell-shaped  form,  a  property  of  the  capac- 
ity  distribution  which  is  frequently  observed. 
A  central  concept  in  the  distribution  approach  is  the  capacity  distribu- 
tion,  +([),  the  distribution  of  capacity  output  in  terms  of  the  labour 
*  This  paper  was  written  with  financial  support  of  the  Netherlands  Organization  for  the 
Advancement  of  Pure  Research  (ZWO).  I would  like  to  thank  Professor  L.  Johansen  and 
Dr.  F.R.  Fsrsund  for  their  comments. 
01651765/83/0000-0000/$03.00  0  1983  North-Holland 218  J.  Muysken  /  Transformed  beta-capacity  dmrributions  o/producl~on  units 
input  coefficient,  5.  ’ Its  normalised  form  is 
&)  =+(5)/J,  t>E”>O  where  J= 
/  -+(W. 
5” 
(1) 
J  is  the  capacity  output  of  the  industry.  Also  normalising  aggregate 
output  and  labour  of  the  industry,  Y and  L,  respectively,  yields  y  =  Y/J 
and  I =  L/J.  Assuming  efficient  behaviour,  the  short-run  production 
function  of  the  industry  is  uniquely  determined  by  the  capacity  distribu- 
tion  and  vice-versa.  This  function,  y  =f(l),  has  the  following  properties 
[Muysken  (1979,  pp.  77-79)]: 
f (0) = 0,  f(l)‘0  for  o-d-d,  f(i)=  1, 
f’(O)=  l/t’,  f’(l)=q  for  O<l<i,  f’(l)=O,  (2) 
f”(l<  0)  for  0 <I  < i, 
where  I!=  L/J,  Lis  the  amount  employed  when  the  industry  is  operating 
at  full  capacity,  and  q  stands  for  the  wage  rate  in  terms  of  output  price. 
Finally  the  capacity  distribution  underlying  a  given  short-run  produc- 
tion  function  is  given  by 
(3) 
where  the  function  d  is  given  by f ‘(1)  =  q  =  I =  d(q). 
2.  Transformed  beta-capacity  distributions  ’ 
The  capacity  distribution  can  always  be  expressed  as  a  function  of 
Then  holds 
~h(~0/5)d(50/~)  =  w(~“/~)d(~o/&  (4) 
Cf.  Johansen  (1972)  and  Sato  (1975).  We  assume  a  putty-clay  model  in  which  production 
units  operate  with  only  two  inputs:  one  variable  input,  labour,  and  one  fixed  input, 
capital.  For  the  putty-putty  case  see  Sato  (1975);  for  the  extension  to  more  variable 
inputs  see  Johansen  (1972)  and  Sato  (1975,  ch.  IV). 
The  idea  to  transform  capacity  distributions  to  beta  functions  is based  on  Sato  (1975,  chs. 
XIV  and  XVII). J.  Muysken  /  Transformed  beta-capacity  distributions  of  production  units  219 
Table  1 
Maximum  and  mean  of  the  (transformed)  B(X,  p)  -  capacity  distributions,  G(x)  and  i(t). 
Maximum 
Mean 
x*=x-l  h+!J 
h+p-2 
[*=---.[O 
A+1 
h  x  =  ~  x+p-1 
XfP 
$=---- 
h-l  .P 
where  w(&‘/t)  is  called  the  transformed  capacity  distribution.  The 
normalized  form  of  this  distribution  is 
4x1  G(x)=y,  o<x=[O/t<  1,  J  = 
/ 
‘w(x)dx. 
0  (5) 
Now  we  are  interested  in  those  cases  where  g(x)  is  a  beta  function,  i.e., 
a(x)  = 
1 
B(X,  P) 
.&‘.(l  -X)“_‘,  o<x<  1,  X,P>O.  (6) 
The  capacity  distribution  corresponding  to  G(X)  is 
(7) 
Table  1 summarises  the  mean  and  the  maximum  for  both  distributions; 
G(x)  and  G(t)  have  a maximum,  and  hence  are  bell-shaped,  when  X >  1, 
p >  1 and  X >  0,  p >  1, respectively. 
The  aggregate  short-run  production  function  corresponding  to  the 
capacity  distribution  (7)  has  the  additional  property  to  eq.  (2)  that  it  can 
be  written  as  an  explicit  function  of  I/i,  3 It  is  impossible,  however,  to 
derive  its  explicit  form  without  further  assumptions.  In  fact,  its  deriva- 
tion  only  is  analytically  manageable  when  we  have  either  p =  1 or  h =  1 
or  2 [Muysken  (198 lb,  p.  5)]. As  we  shall  show  below,  these  are  the  cases 
of  the  Cobb-Douglas  function,  the  CES  function  and  a  VES  function, 
respectively. 
’  Muysken  (1981b,  p.  5).  This  property  allows  one  to  derive  the  long-run  production 
function  as  an  envelope  of  the  short-run  production  functions  with  respect  to  i,  Muysken 
(1981a). 220  J.  Muysken  /  Transformed  beta-capacity  distributions  of production  units 
3.  The  Cobb-Douglas,  the  CES  and  a  VES  production  function  4 
B( B,  1).  Houthakker  (1956)  already  proved  that  the  capacity  distribu- 
tion  underlying  the  Cobb-Douglas  function  is  a  Pareto  distribution, 
which  is  a  special  form  of  the  beta-distribution.  However,  a  well-known 
problem  is  that  one  has  to  assume  5’  =  0  in  order  to  derive  the 
Cobb-Douglas  production  function.  In  the  more  realistic  case  of  to  >  0, 
the  Pareto  capacity  distribution: 
iw =  (@E0)(50/5)P+‘,  P’  1  (8) 
which  is  a  transformed  B(  ,8,  I)-distribution,  yields 
Y=l-{l-1/I}(P/(P-l)).  (9) 
This  short-run  production  function  has  been  studied  in  Kuipers  (1970) 
for  the  case  of  B =  2. 
B(l,  a/(1  -  a)).  Levhari  (1968)  derived  the  capacity  distribution  corre- 
sponding  to  the  CES  function  below: 
y=  [a.fp+p]l’p,  P<O,  O<u=1  1_p<l.  (10) 
The  distribution  is  a  transformed  B(l,  a/(1  -  a))-distribution  and  has 
the  following  form:  5 
0 c  z =  ((O/[)‘_” < 1.  (11) 
This  distribution  is  bell-shaped  for  u >  i;  the  greater  u,  the  flatter  the 
distribution  is  and  the  further  stretched  out  away  from  6’.  A  drawback  is 
that  the  distribution  does  not  converge:  iis  infinite.  Hence  the  amount  of 
4  The  tram-log  production  function  is  inconsistent  with  the  distribution  approach,  Muys- 
ken  (1981b,  sect.  6). 
5  It  is  obvious  that  it  is  necessary  to  transform  the  distribution  to  I  =  (s$“/()’  m0 instead  of 
x  =  .$0/t. .I.  Muysken  /  Transformed  beta-capacity  distributions  of  production  units  221 
labour  employed  in  an  industry  operating  at  full  capacity  is  infinite. 
B(2,  l/(cx  -  1)).  The  production  function: 
y=a,l-b.l”,  Ly>  1  (12) 
is  a  function  with  a  variable  elasticity  of  substitution.  The  case  of  CY  =  2 
has  been  elaborated  in  Kuipers  (1970)  the  case  of  (Y  =  3/2  in  Muysken 
(1979).  The  corresponding  capacity  distribution  is 
(13) 
which  is  a  transformed  B(2,  l/(cx  -  I))-distribution.  The  distribution  is 
bell-shaped  for  1 <  (Y  <  2;  the  lower  (Y, the  flatter  the  distribution  is  and 
the  further  stretched  out  away  from  to. 
References 
Houthakker,  H.S.,  1956,  The  Pareto  distribution  and  the  Cobb-Douglas  production  func- 
tion  in  activity  analysis,  Review  of  Economic  Studies  23,  27-3  1. 
Johansen,  L.,  1972,  Production  functions,  an  integration  of  micro  and  macro,  short  run  and 
long  run  aspects  (North-Holland,  Amsterdam). 
Kuipers,  SK.,  1970,  De  betekenis  van  vraag-  en  aanbodfactoren  in  groeimodellen  met  Cen 
sector,  Doctoral  dissertation  (Groningen). 
Levhari,  D.,  1968,  A  note  on  Houthakker’s  aggregate  production  function  in  a  multi-firm 
industry,  Econometrica  36,  15 I-  154. 
Muysken,  J.,  1979,  Aggregation  of  putty-clay  production  functions,  A  methodological  study 
of  the  distribution  approach,  applied  to  the  Japanese  cotton  spinning  industry,  Doctoral 
dissertation  (Groningen). 
Muysken,  J.,  1981a.  Pareto-family  capacity  distributions  and  aggregate  short-run,  full- 
capacity  and  long-run  production  functions,  in:  E.  Charatsis,  ed.,  Proceedings  of  ESEM 
1979  (Athens)  forthcoming. 
Muysken,  J.,  1981b,  Transformed  beta-capacity  distributions  of  production  units  in  an 
industry,  Memo.  (Institute  of  Economics,  University  of  Oslo,  Oslo). 
Sato,  K.,  1975,  Production  functions  and  aggregation  (North-Holland,  Amsterdam). 

1981b, Transformed beta-capacity distributions of production units in an industry,

A note on Houthakker’s aggregate production function in a multi-firm industry,

Aggregation of putty-clay production functions, A methodological study of the distribution approach, applied to the Japanese cotton spinning industry, Doctoral dissertation

De betekenis van vraag- en aanbodfactoren in groeimodellen met Cen sector, Doctoral dissertation

Pareto-family capacity distributions and aggregate short-run, fullcapacity and long-run production functions,

Production functions and aggregation (North-Holland,

Production functions, an integration of micro and macro, short run and long run aspects (North-Holland,

The Pareto distribution and the Cobb-Douglas production function in activity analysis,

Transformed beta-capacity distribution of production units.

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Transformed beta-capacity distribution of production units.

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