Optimal Advertising And Optimal Quality

OPTIMAL ADVERTISING AND OPTIMAL QUALITY
By

ROBERT

DORFMAN

and

PETER

0. STEINER*

Lawrence Abbott discussed some of the principles of quality competition in a recent issue of this Review.' Most of the conclusions obtained by
Abbott and a number of other results of some interest can be derived
more easily by approaching the problem of differentiated competition
from a broaderpoint of view than Abbott's, using rather simple analytic
tools. We demonstrate the technique which we have in mind, along with
some results of intrinsic interest, by applying the technique to a few
problems including Abbott's.

1. Joint Optimization of Advertising Budget and Price
Theorem: A firm which can influence the demand for its product by
advertising will, in order to maximize itsprofits, choose the advertising
budget and price such that the increase in gross revenue resulting from a
one dollar increase in advertising expenditure is equal to the ordinary
elasticity of demand for the firm's product. The proof of this statement
will be given immediately. As a clarifying preliminary we state that
we mean by advertising any expenditure which influences the shape orposition of a firm's demand curve and which enters the firm's cost function as a fixed cost, i.e., a cost which does not vary with the quantity
of output. This concept corresponds generally, but not exactly, to the
usual concept of advertising. It includes expenditures on billboards,
newspaper space, radio time, interior decoration of a place of business,
air conditioning of sales space, etc. Andnow, the proof.
We consider a firm which makes two kinds of choice: the price of its
product and the amount of its advertising budget. Assuming this to be
so, the relationship between the quantity the firm can sell per unit of
time, q, its price, p, and its advertising budget, s, can be denoted by the
formula
q = f(p, s).

We assume thatf(p, s) is continuous and differentiable.
In order todetermine the optimal price-quantity-advertising

(1)

con-

* The authors are associate professor and assistant professor of economics, respectively, at
the University of California, Berkeley.
1 Lawrence Abbott, "Vertical Equilibrium under Pure Quality Competition," Am. Econ.
Rev., Dec. 1953, XLIII, 826-45.

DORFNIAN AND STEINER: OPTIMAL ADVERTISING AND QUALITY 827

stellation it isconvenient for expository reasons to analyze the situation
in two steps. In the first step we regard the quantity of output as fixed
and specify the optimal price-advertising constellation for selling that
predetermined quantity. Then we let quantity vary and seek its optimum. The advantage of this procedure is that cost considerations,
other than the cost of advertising, do not enter thefirst step.2
Suppose, then, that price be changed by a small amount, dp, and
advertising expenditure by a small amount, ds. The change in the level
of sales will be the total differential of equation (1) or:
dq

dp +

--

ap

os

ds.

In order for quantity not to change as a result of these variations, dp
and ds must be chosen in such a way that they have equal and opposite
effectson quantity; so that dq = 0. That is,
0]

dp

--

os

ds, assuming

dfpop

As
-

#

0.

(2)

op
The result of these variations is to change gross revenue by the amount
qdp, change advertising expenditure by ds, and leave the volume of sales
and aggregate production cost unchanged. The net effect on profit is,
therefore:

qdpp-ds-

q

As

if + 1 ds.

ap

(3)j

Now we must consider two cases: first, where the original level of
advertising from which the variations were measured was positive, and
second, where the original level of advertising was zero. We now show
that a positive level of advertising cannot be optimal unless the
quantity in parentheses in equation (3) is zero. For, if that quantity
were positive we could choose a negative...