Math 132
Lesson 3: Price Data for Two Markets and Linear
Estimation
Application
This lesson is a continuation of Lesson 2: Price Data and Trendlines, where we considered the profit
from one market with price per unit function that depended only on the sales in a single market. In
this lesson we will consider the two market problem, where the prices per unit in each market
depend on sales in both markets. The math model for the price data will be linear in each variable for
the sales. However, the resulting algebraic problem is a little more difficult, and so, we will use the
Excel command called LINEST for linear estimation of the price data. We will indicate how this linear
price function of the two sales variables can be used to determine the maximum profit.
Math Model
Let the number of units sold in market A be x
A
, and the number of units sold in market B be x
B
.
Consider a two market problem with the following price per unit functions:
p
A
= 97 - x
A
/10 - x
B
/200
and
p
B
= 83 - x
B
/20 - x
A
/100.
Notice that in each market, the price starts high. The price in each market is reduced a certain
amount by sales in its market, and also reduced (a smaller amount) by sales in the other market. The
new terms are y/200 and x/100, and they represent some interaction between the two markets.
Where would these functions have come from? The coefficients 97, -1/10 and -1/200, for the price in
market A, and the corresponding coefficients for the price in market B, must be determined from
previous sales data. If the data is approximately linear in each variable while the other variable stays
fixed, then a linear function of 2 variables will be a good model
for the price in each market:
p
A
=
I
A
−
r
AA
x
A
−
r
AB
x
B
p
B
=
I
B
−
r
BA
x
A
−
r
BB
x
B
where
variable in
our model
definition
x
A
sales level in market A
x
B
sales level in market B
p
A
price in market A
p
B
price in market B
I
A
initial price in market A
I
B
initial price in market B
r
AA
price reduction in market A per suit sold in market A
r
AB
price reduction in market A per suit sold in market B
r
BA
price reduction in market B per suit sold in market A
r
BB
price reduction in market B per suit sold in market B
It is very good practice to make a
complete table of all your
symbols and their definitions in
your mathematical model. It
helps a great deal in avoiding
confusion and searching through
lots of pages.

Once the two price functions p
A
and
p
B
are found, the profit as a function of two variables can be
determined. Then the revenue is