The Camera Shop sells two popular models of digital cameras. The sales of these

Question

The Camera Shop sells two popular models of digital cameras. The sales of these products are not independent of each other, but rather if the price of one increase, the sales of the other will increase. In economics, these two camera models are called substitutable products. The store wishes to establish a pricing policy to maximize revenue from these products. A study of price and sales data shows the following relationships between the quantity sold (N) and prices (P) of each model:

   NA = 190 - 0.7PA + 0.35PB

   NB = 300 + 0.07PA - 0.6PB

Construct a model for the total revenue and implement it on a spreadsheet. Develop two-way data table to estimate the optimal prices for each product in order to maximize the total revenue. Vary each price from $250 to $500 in increments of $10.

Max profit occurs at Camera A price of $ .

Max profit occurs at Camera B price of $ .

Explanation / Answer

NA = 190 - 0.7PA + 0.35PB

   NB = 300 + 0.07PA - 0.6PB

where N is the quantity sold and P the price

Revenue from car A = NA*PA = 190PA-0.7PA2+0.35PAPB

Revenue from car B = 300PB + 0.07PAPB - 0.6PB2

Total revenue R =  190PA+300PB-0.7PA2-0.6PB2+0.42PAPB

Let PA,PB be denoted by x and y

R(x,y) = 190x+300y-0.7x2-0.6y2+0.42xy

x varies from 250 to 500 and y 250 to 500

Maximum when both are equal to 500

Using partial derivatives

R(x,y) = 190x+300y-0.7x2-0.6y2+0.42xy

Rx = 190-0.14x+0.42y

Ry = 300-0.12y+0.42x

Set both to 0 and solve for x and y

Multiply I equation by 3

570-0.42x+1.26y =0 and

300+0.42x-0.12y=0

Adding 870 = 1.14y

y = 763.14

300-0.12(763.14)+0.42x=0

x = 496.24

i.e. when x = 496 and y =763 revenue is maximum.

Max profit occurs at Camera A price of $ 496

Max profit occurs at Camera B price of $ 763

R(x) 190x+300y-0.7x^2-0.7y^2+0.42xy x y R 250 250 106875 260 260 110500 270 270 114075 280 280 117600 290 290 121075 300 300 124500 310 310 127875 320 320 131200 330 330 134475 340 340 137700 350 350 140875 360 360 144000 370 370 147075 380 380 150100 390 390 153075 400 400 156000 410 410 158875 420 420 161700 430 430 164475 440 440 167200 450 450 169875 460 460 172500 470 470 175075 480 480 177600 490 490 180075 500 500 182500 500 250 167500 250 500 95625 300 450 120900

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