Page 36 - Microeconomics, Fourth Edition

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10 CHAPTER 1 ANALYZING ECONOMIC PROBLEMS
TABLE 1.1 New Beer Sales Resulting from Amounts Spent on TV and
Radio Advertising
New Beer Sales Generated
(in barrels per year)
Total Spent TV Radio
$ 0 0 0
$ 100,000 4,750 950
$ 200,000 9,000 1,800
$ 300,000 12,750 2,550
$ 400,000 16,000 3,200
$ 500,000 18,750 3,750
$ 600,000 21,000 4,200
$ 700,000 22,750 4,550
$ 800,000 24,000 4,800
$ 900,000 24,750 4,950
$1,000,000 25,000 5,000

In light of the information in Table 1.1, how would you allocate your advertising
budget if your objective is to maximize the new sales of beer?
This is a constrained optimization problem. You want to allocate spending on
TV and radio in a way that maximizes an objective (new sales of beer) subject to the
constraint that the total amount spent on TV and radio must not exceed your
$1 million advertising budget. Using notation similar to that introduced in the pre-
vious section, if B(T, R) represents the amount of new beer sales when you spend
T dollars on television advertising and R dollars on radio advertising, your con-
strained optimization problem is

max B(T, R)
(T,R)
subject to: T R 1 million

A quick reading of Table 1.1 might suggest an “obvious” answer to this problem:
Allocate your entire $1 million budget to TV spots and spend nothing on radio. After
all, as Table 1.1 suggests, a given amount of money spent on TV always generates
more new sales than the same amount of money spent on radio advertising. (In fact,
a given amount of TV advertising is five times as productive in generating new sales
as is the same amount of radio advertising.) However, this answer is incorrect. And the
reason that it is incorrect illustrates the power and importance of constrained opti-
mization analysis in economics.
Suppose you contemplate spending your entire budget on TV ads. Under that
plan, you would expect to get 25,000 barrels of new sales. But consider, now, what
would happen if you spent only $900,000 on TV ads and $100,000 on radio ads. From
Table 1.1, we see that your TV ads would then generate 24,750 barrels of new beer
sales, and your radio ads would generate 950 barrels of new beer sales. Thus, under
this plan your $1 million budget generates new beer sales equal to 25,700 barrels. This
is 700 barrels higher than before. In fact, you can do even better. By spending
$800,000 on TV and $200,000 on radio, you can generate 25,800 barrels of new beer
sales. Even though Table 1.1 seems to imply that radio ads are far less powerful than