Answer to Question #159055 in Economics of Enterprise for tuba

Question

Sales are the function of advertising in The Dawn and Diva Magazine (X, Y). 

S = XY2

If the price of advertising in The Dawn and Diva Magazine is Rs.5 and Rs.10 respectively. The total budget for advertising is Rs.105. For maximizing the sales of Dawn and Diva Magazine find out the best combination of advertisement in newspapers and magazines by using Lagrangian multiplier

Answer 1

The sales function is S = XY²

By a positive monotonic transformation, we get

lnS = lnX + 2lnY

Given:

Px = 5

Py= 10

M = 105

M ≥ PxX + PyY

105 ≥ 5X + 10 Y

Forming the Lagrangian, we get:

α = lnX + 2lnY + λ(105 – 5X -10Y)

λ is the Lagrangian multiplier

$\frac{∂α}{∂X} = \frac{1}{X}-5λ \\ 5λX = 1 \\ \frac{∂α}{∂Y} = \frac{2}{Y}-10λ \\ 5λY = 1 \\ \frac{∂α}{∂λ} = 105 - 5X -10Y = 0 \\ \frac{5X}{5Y}= 1 \\ X = Y \\ 105 = 5X + 10Y$

X* = Y* = 7