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Marketing Notes – Quantitative Methods Review

Performance Maximization – Optimal Sales Goal

Given the following, how do we maximize performance?
Revenue per unit is $3.00
Fixed costs are $2000
Variable cost of $.0002 times Quantity2

Use of the Calculus

If the budget is $10,000, what are the optimal allocations for advertising and promotion?
A sales manager estimates that monthly sales can be represented by

S = 60,000 + 3A – 0.00025A2 + 6P – 0.004P2

Where:
S = sales in units
A = advertising in dollars
P = promotion activity in dollars

If the budget is $10,000, what are the optimal allocations for advertising and promotion?

Constrained Optimization

Constrained problem in multiple variables:
Max S = 60,000 + 3A – 0.00025A2 + 6P – 0.004P2
s.t. A + P = 10,000

Use of Lagrange Multipliers
Optimal solution:
A* = $5923, P* = $4077, S* = 26,973
Sales function is concave so we have a maximum.

Breakeven Analysis

The prime condition is:
Revenue = Total Cost
Price * Quantity = Fixed Cost + (Variable Cost) * Quantity
BEQ = FC ÷ (price – VC)

Important Breakeven Concepts

Example:
Total Cost = $42,000 + $8.4 * Q
Revenue = $12 * Q
Breakeven condition:
Revenue – Total Cost = 0
$12 * Q – $42,000 – $8.4 * Q = 0
BEQ = 11,667 units

Example:
Your department is “brainstorming” for ideas to meet earnings goals imposed by higher management. One idea sounds good, but you remember the corporate planning staff reviews your ideas using terms like net present value, contribution, etc.

Available Information

How many units must be sold to break even?

Solution:
BEV table

BEQ = FC ÷ Contribution = 280,000 ÷ 0.75 = 373,334

Including a Target:

Solution:

BEQ = (FC + Target) ÷ Contribution
= (FC + 0.25(FC)) ÷ 0.75
= (280,000 * 1.25) ÷ 0.75 = 466,667

Including an After Tax Target:

Solution:
BEQ = (FC + After Tax Target) ÷ Contribution
= (FC + (0.25*FC) ÷ (1 – MTR)) ÷ 0.75
= (280,000 (1 + 0.25 ÷ 0.6)) ÷ 0.75
= 529,000

Which Approach is Better?