This is a comment on an article on the revenue maximization hypothesis by William BaumolOligopoly; revenue maximization

De Bartolo, Gilbert

Haveman, Robert

English

Research Papers in Economics

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The Revenue Maximization Oligopoly Model: Comment
Author(s): Robert Haveman and Gilbert De Bartolo
Source: The American Economic Review, Vol. 58, No. 5, (Dec., 1968), pp. 1355-1358
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1814035
Accessed: 06/08/2008 15:22
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The  Revenue  Maximization  Oligopoly  Model:  Comment 
Since Professor Baumol's substitution  of constrained revenue maximiza- 
tion for profit maximization in the objective function attributed to rational 
oligopolists,  the  nature  of  the  relationships  between  advertising  outlay, 
price, output, and production cost in the oligopoly equilibrium solution has 
been radically altered [1]. However, because these relationships are clearly 
articulated  neither  graphically  nor  in  the  mathematical  appendix  of 
Baumol's  book  (even  the  revised  edition),  their  appreciation  has  been 
limited. Moreover, a recent article integrating both the price and advertis- 
ing  outlay  variables  and  touted  as the general statement  of  the  Baumol 
model is,  as will be shown,  but  a special case of the  Baumol  model [2]. 
In this commentary, we shall present a general model of the sales maximi- 
zation hypothesis,  articulating the above relationships and remedying the 
shortfalls in generality. 
In the text of Business Behavior, Value and Growth,  Baumol presents his 
admittedly  "partial"  sales  maximization  argument  together  with  its 
equilibrium advertising outlay-price-output  solution. The equilibrium out- 
put is defined with production costs and price as variables but with adver- 
tising  outlay  excluded.  The  equilibrium level  of  advertising  depends  on 
advertising  outlay  with  production  costs  and  presumably  output  held 
constant.  Total  revenue is a standard parabolic function  of output,  mar- 
ginal production cost is a positive  and increasing function  of output,  and 
marginal revenue is a positive  but decreasing function  of advertising out- 
lay. Given a minimum profit constraint, Baumol demonstrates the impossi- 
bility of any combination of advertising outlay and output yielding both a 
profit above the constraint and maximum sales revenue [1, p. 60]. 
In the mathematical note, this problem is stated as one of maximizing a 
total revenue function  (price times quantity)  subject to a minimum profit 
constraint which includes both  advertising and production outlays.  Here, 
marginal production cost is taken to be positive and relevant to the deter- 
mination of the equilibrium price and output [1, p. 69]. 
In an article by R. L. Sandmeyer [2], Baumol is criticized for failing to 
integrate the sales maximizer's price policy with his advertising policy in 
describing the  equilibrium. This  situation  is  then  "remedied" through a 
graphical model which "allows price, advertising outlay,  and product out- 
put to change" [2, p. 1078]. Beginning in a no-advertising situation in which 
total revenue is maximized at an output unconstrained by the profit mini- 
mum, Sandmeyer demonstrates that profit in excess of the minimum will be 
devoted  to advertising outlay,  generating a new and higher total  revenue 
function, a new price and a new excess profit level which in turn is devoted 
to advertising outlay,  etc., until eventually  equilibrium output,  price, and 
advertising outlay  are determined. At this equilibrium, the level  of profit 
equals the minimum acceptable. 
Two  things  should  be  noted  with  respect  to  this  model.  First,  it  is  a 
"partial" model in that  all production and marketing costs are fixed and, 
consequently,  the nonadvertising  marginal cost  of increasing sales is zero 1356  THE AMERICAN  ECONOMIC  REVIEW 
over the relevant range. Second, because of the implicit zero marginal cost, 
the equilibrium defined by Sandmeyer occurs at the peak of one of the total 
revenue functions,  i.e.,  where marginal revenue is zero-a  solution which 
is inconsistent with the basic equilibrium conditions of the model as stated 
by Baumol [1, pp. 61, 69]. 
In the following model, the equilibrium for a revenue maximizer will be 
shown when price, cost, output, and advertising outlay are all free to vary. 
This model will be referred to as the general Baumol model of which both 
Baumol's "partial" presentation and Sandmeyer's model are special cases. 
Define,  as does Baumol, 
D(x,a)-demand  function representing the price which will be paid when 
x units are offered after a dollars of advertising outlay; 
ir-minimum  acceptable profits; 
C(x)-total  cost of production function: 
A (a)-total  cost of advertising function; 
x  D(x,a)-total  revenue function representing the maximum total revenue 
saved from selling x units after a units of advertising. 
Given these definitions, we maximize the  total  revenue function,  subject 
to a profit constraint: 
(1)  x-D(x,a)  -7r-  C(x)  -  A(a)  >0. 
Using  the  Lagrange multiplier technique  (and Kuhn-Tucker conditions), 
we have: 
(2)  -{  x - D(x,  a)-\,[x  - D(x,  a)  -Xr-  C(x)  -A  (a)]  <  O, 
Ox 
da  (3)  aa  {x.D(x,  a)  -X[x.D(x,  a)  -7  -  C(x)  -  A(a)]}  <  0. 
Differentiating  (2)  and (3),  solving for X, and substituting,  we  have,  for 
interior points, 
dD  dC  dA  AD  dA 
(4)  x-*-  =  D-d  +  x-*-, 
cla  dx  da  Ox  da 
which  is  the  statement  of  the  equilibrium  condition.  Letting  TR(x,a) 
=x-D(x,a),  this equilibrium condition (4) can be written as:' 
dTR (dC  dTR (dA 
(Oa  dx  Ox  da 
Note  that,  by assumption, the Baumol formulation requires the following 
conditions, 
aTR  dC  dA 
-  >0,  -> -  ,  and  x-  =  1, 
da  dx  da 
1 This condition  can easily  be generalized  to allow  for more  than  one output  and advertisig 
outlay. See [1, pp. 68-69]. COMMUNICATIONS  1357 
Total 
Revenue  and 
Cost 
fl+C+a+l  _ 
TR(A.a+l) 
T 
(x)+a 
Tr+C+a  Tf  __  T1+ 
-rr+c 
lT4a+l 
0~~~~~~~~~~~~ 
FIGuRE 1 
[1, pp. 54, 59] which, in  turn, necessitate  aTRIax>O.  This is to be com- 
pared with the aTRIax = 0 claimed by Sandmeyer as a characteristic of the 
equilibrium solution. 
Graphically, Sandmeyer's equilibrium position is shown as To in Figure 1. 
With  the  TR functions  representing the  relationship of  total  revenue to 
output  at  different levels  of advertising outlay,  the  minimum profit con- 
straint represented by 7r,  and the production cost function implied by the 
horizontal curve 7r+C, this solution gives an output of OQo,  a price of ToQo 
/OQo, and advertising outlay  of coTo=aobo. Clearly, this  solution implies 
aTR/ax=dC/dx=0,  which,  as  noted  above,  is  inconsistent  with  the  re- 
quirements of the generalized Baumol model. 
The  solution  of the general Baumol model is  also  shown  in  Figure  1. 
Again, the  TR functions  represent a family  of total  revenue functions  at 
different levels  of  advertising  outlay  and the  level  of  minimum profit is 
given by 7r.  In this case the production cost function is implied by 7r+C(x) 
+a.  For each level of advertising outlay, a new total cost function (includ- 
ing 7r) is  observed.  At  output  OQo, the  advertising  outlay  is  aobo  =  coTo, 
total  production cost is equal to boTo,  and total  revenue equals total  cost 
including the minimum profit constraint. These  conditions, then  are true 
of both the Sandmeyer model and the general Baumol model at output Qo. 
However,  while the sales maximizer is in equilibrium at Qo in  the  Sand- 
meyer model, he is not in equilibrium in the  general Baumol model. The 
next unit of advertising will be purchased, output will be cut back from OQo 
to OQj,  the price of the output will be raised from ToQo/OQo  to  T1Ql/OQ1, 
and total production cost will decrease from Tobo  to T1b1.  The change will 
be undertaken because total sales revenue is increased from ToQo  to  T1Qj. 
In general, then,  the sales maximizer in the general Baumol model will 1358  THE AMERICAN  ECONOMIC  REVIEW 
demonstrate a smaller output,  a greater advertising outlay,  a higher price, 
and a greater sales volume than will the sales maximizer in the Sandmeyer 
version of the Baumol model. With dC/dx>  0, the oligopolist will find it in his 
interest  to  allocate  profit surplus from production  to  advertising in  the 
process  of  generating  sales  revenue.  Moreover,  in  the  general  Baumol 
model the tendency  toward high product price, excessive advertising out- 
lay,  and more intense  output  restriction will be positively  related to  the 
elasticity  of marginal cost. 
ROBERT  HAVEMAN  AND  GILBERT  DE  BARTOLO* 
* The authors  are,  respectively,  associate  professor  of economics,  Grinnell  College  and senior 
economics  student,  Grinnell  College. 
REFERENCES 
1.  W. J. BAUMOL, Business Behavior, Value and Growth,  rev. ed. New York 
1967. 
2.  R. L.  SANDMEYER,  "Baumol's Sales-Maximization  Model:  Comment," 
Am. Econ. Rev., Dec.  1964, 54, 1073-1081. 
Incidence  of  the  Corporation Income  Tax  in 
U.S.  Manufacturing:  Comment 
In  responding to previous critiques of our incidence study  [41, we have 
conceded that our initial effort should eventually  come to be replaced by a 
more complex approach, involving a structural model in which price, wage, 
and shifting behavior are specified, and all equations are identified. While 
the required data are exceedingly difficult to obtain, such a reformulation 
may eventually be forthcoming. Such a model may not only provide better 
information on total shifting, but also show how shifting comes about and 
what "direction" it takes. It is with great anticipation, therefore, that one 
follows beyond the opening pages of a recent article by R. J. Gordon, where 
such an approach appears to be taken.'  But  as the paper proceeds, it  be- 
comes apparent that no such improvement is offered. 
Tax-less Model 
Gordon begins with  a model for a tax-less world. He  defines profits as 
sales minus costs,  with  costs  a function  of quantity,  wage rates, and ma- 
terial  prices.  Sales  equals  product  price  times  quantity,  where  price  is 
determined  by  a  behavioral  equation  involving  mark-up  prices.  This 
assumption  is  worth  testing,  although  other  behavior  hypotheses  might 
have  been used as well. In  equation  (7),  Gordon attempts  to present the 
system in reduced form. Some of the variables in (7) being unavailable, he 
proceeds  to  make  certain  substitutions,  thus  arriving at  equation  (12), 
1 See [21.  To simplify  and to permit  reference  to our  model,  we consider  the Gordon  model 
only  in its rate of return  formulation. 

The Revenue Maximization Oligopoly Model: Comment

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