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 xA, and the number of units sold in market B be xB. Consider a two market problem with the following price per unit functions: pA = 97 - xA/10 - xB/200 and pB = 83 - xB/20 - xA/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: pA=IA−rAAxA−rABxBpB=IB−rBAxA−rBBxBwhere variable in our model definition xAsales level in market A xBsales level in market B pAprice in market A pBprice in market B IAinitial price in market A IBinitial price in market B rAAprice reduction in market A per suit sold in market A rABprice reduction in market A per suit sold in market B rBAprice reduction in market B per suit sold in market A rBBprice 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 pA andpB are found, the profit as a function of two variables can be determined. Then the revenue is