Vande Kamp and Kaiser
Optimal Temporal Milk Advertising 281
and demand, results were also obtained when starting values in the actual advertising pol-
the upper bound on monthly advertising was icy. The actual or historical advertising pol-
removed.6 The optimal advertising policies icy is shown in figure 1. One can observe
for these four optimizations were compared that the actual advertising exhibits some puls-
to the results from two additional advertis- ing, especially between months 73 and 98
(January 1992–January 1994) where a pattern
of two months of low advertising followed by
one month of high advertising is prevalent.
The first optimization (figure 2) shows the
optimal advertising policy when maximum
advertising was two times the historical aver-
age advertising level (a¯ = $1ꢀ 702). A clear
limit cycle emerges where, after the initial
convergence from the starting values, the
optimal policy is three months of zero adver-
tising followed by three months of advertis-
ing at the maximum level, a¯. Convergence
to this pulsing strategy is relatively short—by
the twenty-seventh month, the optimal adver-
tising pattern has already emerged.
The second and third optimal advertising
policies, which are shown in figures 3 and 4,
exhibit similar pulsing patterns. In the second
optimal advertising policy, where the maxi-
mum advertising level was $2,128, the stable
pattern includes four months of zero adver-
tising followed by two months of advertising
at a¯, followed by one month of advertising
at $1,731. The third optimal advertising pol-
icy, where the maximum advertising level
was $2,553, consists of two alternating seven-
month cycles. The stable pulsing pattern
shows four months of zero advertising fol-
lowed by two months of advertising at a¯,
followed by one month of advertising at zero
or $1,769, depending on the cycle.
ing policies. These two include a uniform
advertising policy, where advertising was held
constant at b every month, and the actual
advertising policy for the period January 1986
to June 1995. It is important to note that the
discounted sum of real advertising expendi-
tures over time is the same for all six adver-
tising policies.
Careful analyses should evaluate the sen-
sitivity of the optimal advertising policies
to any assumptions. Therefore, sensitivity of
the results to demand seasonality, changes in
the advertising grid, and interest rates were
investigated. Also several additional opti-
mizations were performed to evaluate the
impact of changes in asymmetry on the opti-
mal advertising policy. These exercises are
discussed in detail in the results.
Fluid milk advertising from July 1994 to
June 1995 (denoted as periods one through
twelve in the optimizations) was used as the
starting values for advertising, a12. Also, the
starting value for savings, that is, savings in
period one, was set to zero (s1 = 0). For
each optimization, the optimal policy function
was used to determine the optimal fluid milk
advertising and the corresponding fluid milk
demand for 100 months beginning in month
thirteen. Using the same starting values, fluid
milk demand was also determined for the uni-
form and the actual advertising policies.
In the fourth optimal advertising pol-
icy, where the upper bound on monthly
advertising was removed, the optimal puls-
ing pattern consists of five months of zero
advertising followed by one month of adver-
tising at $5,138. While this result may not
be unrealistic, it should be treated with
caution since the pulse of advertising is sig-
nificantly outside the range of the data, and
the error associated with the corresponding
fluid milk demand may be very large. More-
over, as discussed earlier, large purchases of
advertising during the targeted time slots in
one month could potentially drive up the
price of advertising.
The optimal advertising policies show
advertising to be a periodic function of time;
i.e., a(t) = a(t − k) for a given period (length
of cycle in periodic function) k and for all
months t after convergence. While the total
cycle length for the first optimal policy was
Results
Optimal Advertising Policies
In the uniform advertising case, advertising
was constant at $851 per million people per
day after the initial twelve months of start-
ing values. The average daily advertising for
January 1986 to June 1995 was used after the
6 The grid of savings for the first three optimizations was six
equidistant points between and including zero and $5,106. Tests
confirmed that the choice of the maximum savings grid point
did not impact results. Similarly, the grid of monthly advertising
consists of five equidistant points between and including zero
and the maximum advertising level. For example, for the second
optimization, the advertising grid was $0, $532, $1,064, $1,596, and
$2,128. For the fourth optimization, the grid of savings was six
equidistant points between and including zero and $6,808. The
grid of advertising was five equidistant points also between and
including zero and $6,808.