Session 08-03 - Cost Analysis & Pricing Decisions

Section 08: Financial Mathematics

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz - 10 Minutes

Quick Review from Session 08-02

Test your understanding of Annuities

Calculate the future value of saving 250/month for 15 years at 4% annual interest.
Find the monthly payment for a 30,000 loan at 5.4% for 4 years.
After 12 payments on a 15,000 loan at 6% annual for 3 years, what is the outstanding balance?
What is the present value of receiving 1,500/month for 10 years at 6% annual?

Learning Objectives

What You’ll Master Today

Understand cost function components: fixed vs. variable costs
Calculate variable cost per unit \(k_v(x)\)
Find the short-term lower limit price (Kurzfristige Preisuntergrenze)
Find the long-term lower limit price (Langfristige Preisuntergrenze)
Make pricing and production decisions using cost analysis

. . .

Exam-Critical Topic

These pricing concepts appear frequently on the Feststellungsprufung!

Part A: Cost Function Review

Components of Total Cost

Every business has two types of costs:

. . .

Cost Components

Fixed Costs (\(K_f\) or \(FC\)): Costs that don’t change with production quantity

Rent, insurance, salaries, equipment depreciation

Variable Costs (\(K_v(x)\) or \(VC(x)\)): Costs that change with quantity produced

Raw materials, energy, direct labor, packaging

. . .

\[K(x) = K_f + K_v(x)\]

Total Cost Function Examples

Common cost function forms:

. . .

Linear: \(K(x) = 500 + 3x\)

Fixed cost: 500
Variable cost per unit: 3 (constant)

. . .

Quadratic: \(K(x) = 400 + 10x + 0.02x^2\)

Fixed cost: 400
Variable costs increase with quantity

. . .

Cubic: \(K(x) = 1000 + 50x - 0.5x^2 + 0.01x^3\)

Most realistic: economies then diseconomies of scale

Visualizing Cost Functions

Part B: Average Cost Functions

Average Total Cost

How much does each unit cost on average?

. . .

Average Cost (Durchschnittskosten)

\[\bar{K}(x) = \frac{K(x)}{x} = \frac{K_f + K_v(x)}{x} = \frac{K_f}{x} + \frac{K_v(x)}{x}\]

. . .

This can be written as: \[\bar{K}(x) = \underbrace{\frac{K_f}{x}}_{\text{Avg. Fixed Cost}} + \underbrace{\frac{K_v(x)}{x}}_{\text{Avg. Variable Cost}}\]

Variable Cost Per Unit

Variable Cost Per Unit (Stuckkosten variabel)

\[k_v(x) = \frac{K_v(x)}{x} = \frac{K(x) - K_f}{x}\]

This is the average variable cost - what each unit costs in variable expenses alone.

. . .

Example: \(K(x) = 500 + 10x + 0.01x^2\)

\[k_v(x) = \frac{(500 + 10x + 0.01x^2) - 500}{x} = \frac{10x + 0.01x^2}{x} = 10 + 0.01x\]

Why Variable Cost Per Unit Matters

Critical insight for pricing decisions:

. . .

If price \(>\) average total cost: Profit
If price \(<\) average total cost but \(>\) variable cost per unit: Loss, but covering some fixed costs
If price \(<\) variable cost per unit: Should NOT produce - losing money on every unit!

. . .

The variable cost per unit is the absolute minimum you should charge!

Part C: Short-Term Lower Limit Price

What is the Short-Term Lower Limit?

Short-Term Lower Limit Price (Kurzfristige Preisuntergrenze)

The minimum price at which a company should continue production in the short term.

\[p_{min,short} = \min\{k_v(x)\} = \text{minimum variable cost per unit}\]

At this price, the company covers its variable costs but not fixed costs.

. . .

When to use:

Fixed costs are “sunk” (already paid/committed)
Short-term decision: continue or stop production?

Finding the Minimum Variable Cost Per Unit

Method: Use calculus to find the minimum of \(k_v(x)\)

. . .

Step 1: Calculate \(k_v(x) = \frac{K(x) - K_f}{x}\)

Step 2: Find \(k_v'(x)\) and set equal to zero

Step 3: Solve for \(x\)

Step 4: Calculate \(k_v(x)\) at this value

. . .

Alternatively, use: \(k_v'(x) = \frac{K'(x) \cdot x - (K(x) - K_f)}{x^2}\)

Setting this to zero: \(K'(x) \cdot x = K(x) - K_f\)

Example: Finding Short-Term Lower Limit

Given: \(K(x) = 800 + 20x - 0.2x^2 + 0.002x^3\)

Step 1: Variable cost function \[K_v(x) = 20x - 0.2x^2 + 0.002x^3\]

Step 2: Variable cost per unit \[k_v(x) = \frac{20x - 0.2x^2 + 0.002x^3}{x} = 20 - 0.2x + 0.002x^2\]

Example (continued)

Step 3: Find minimum \[k_v'(x) = -0.2 + 0.004x = 0\] \[x = 50\]

. . .

Step 4: Calculate minimum variable cost per unit \[k_v(50) = 20 - 0.2(50) + 0.002(50)^2 = 20 - 10 + 5 = 15\]

. . .

Short-term lower limit price = 15 Euro per unit

The company should not sell below 15 Euro, even in the short term!

Graphical Representation

Break - 10 Minutes

Part D: Long-Term Lower Limit Price

Long-Term Lower Limit Price

Long-Term Lower Limit Price (Langfristige Preisuntergrenze)

The minimum price for sustainable long-term production.

\[p_{min,long} = \min\{\bar{K}(x)\} = \text{minimum average total cost}\]

At this price, the company covers ALL costs (both fixed and variable).

. . .

When to use:

Long-term strategic pricing decisions
Deciding whether to enter/exit a market

Finding the Long-Term Lower Limit

Same method, but for average TOTAL cost:

. . .

Step 1: Calculate \(\bar{K}(x) = \frac{K(x)}{x}\)

Step 2: Find \(\bar{K}'(x)\) and set equal to zero

Step 3: Solve for \(x\)

Step 4: Calculate \(\bar{K}(x)\) at this value

Example: Long-Term Lower Limit

Same cost function: \(K(x) = 800 + 20x - 0.2x^2 + 0.002x^3\)

Step 1: Average total cost \[\bar{K}(x) = \frac{800 + 20x - 0.2x^2 + 0.002x^3}{x} = \frac{800}{x} + 20 - 0.2x + 0.002x^2\]

. . .

Step 2: Find minimum \[\bar{K}'(x) = -\frac{800}{x^2} - 0.2 + 0.004x = 0\]

. . .

Multiply by \(x^2\): \[-800 - 0.2x^2 + 0.004x^3 = 0\] \[0.004x^3 - 0.2x^2 - 800 = 0\]

Example (continued)

Solving \(0.004x^3 - 0.2x^2 - 800 = 0\) (or \(x^3 - 50x^2 - 200000 = 0\)):

Using numerical methods or calculator: \(x \approx 76.4\)

. . .

Step 4: Calculate minimum average cost \[\bar{K}(76.4) = \frac{800}{76.4} + 20 - 0.2(76.4) + 0.002(76.4)^2 \approx 17.44\]

. . .

Long-term lower limit price = 17.44 Euro per unit

For sustainable production, the company must charge at least 17.44 Euro!

Comparing Short-Term and Long-Term

Part E: Business Interpretation

Pricing Decision Framework

Three Pricing Zones

Price Level	Decision
\(p < k_v^{min}\)	STOP production - losing on every unit
\(k_v^{min} \leq p < \bar{K}^{min}\)	Continue short-term - covers variable costs, contributes to fixed
\(p \geq \bar{K}^{min}\)	Sustainable - covers all costs

When to Use Each Limit

Short-term lower limit price:

Factory/equipment already exists
Fixed costs are committed (“sunk”)
Deciding whether to accept a special order
Economic downturn - better to produce and lose less than to stop

. . .

Long-term lower limit price:

Deciding whether to enter a new market
Setting regular prices for new products
Strategic planning and investment decisions

Example: Special Order Decision

A company has excess capacity. A customer offers to buy 100 units at 16 Euro each.

Using our cost function with:

Short-term lower limit: 15 Euro
Long-term lower limit: 17.44 Euro

. . .

Analysis:

16 > 15 (short-term limit) \(\checkmark\)
16 < 17.44 (long-term limit)

. . .

Decision: Accept the order!

While 16 < 17.44 means we’re not covering all costs, each unit sold at 16 contributes 1 Euro toward fixed costs. Better than nothing!

Contribution Margin

Contribution Margin (Deckungsbeitrag)

\[\text{Contribution per unit} = p - k_v(x)\]

This is how much each unit “contributes” toward covering fixed costs.

. . .

At optimal production level (x = 50):

Price: 16 Euro
Variable cost per unit: 15 Euro
Contribution margin: 16 - 15 = 1 Euro per unit

For 100 units: 100 Euro contribution toward fixed costs!

Guided Practice - 15 Minutes

Practice Problems

Work in pairs

Problem 1: Given \(K(x) = 1200 + 30x + 0.1x^2\)

Find the variable cost per unit function \(k_v(x)\)
Find the short-term lower limit price
Find the long-term lower limit price

Problem 2: A company has cost function \(K(x) = 500 + 15x\)

What is the short-term lower limit price?
What is the long-term lower limit price at production of 100 units?
Should they accept an order for 50 units at 18 Euro each?

Part F: Exam-Style Problems

Typical Exam Problem Structure

FSP exam problems often combine these elements:

Given a cost function \(K(x)\)
Find the break-even point(s) given price \(p\)
Find the profit-maximizing quantity
Find the short-term/long-term lower limit price
Interpret the results economically

. . .

Make sure you can distinguish between these questions:

“Kurzfristige Preisuntergrenze” = Short-term = min of \(k_v(x)\)
“Langfristige Preisuntergrenze” = Long-term = min of \(\bar{K}(x)\)

Summary of Key Formulas

Essential Formulas for Cost Analysis

Concept	Formula
Total Cost	\(K(x) = K_f + K_v(x)\)
Variable Cost per Unit	\(k_v(x) = \frac{K(x) - K_f}{x}\)
Average Total Cost	\(\bar{K}(x) = \frac{K(x)}{x}\)
Short-term Limit	\(\min\{k_v(x)\}\)
Long-term Limit	\(\min\{\bar{K}(x)\}\)
Contribution Margin	\(p - k_v(x)\)

Wrap-Up & Key Takeaways

Today’s Essential Concepts

Variable cost per unit \(k_v(x)\) = variable costs divided by quantity
Short-term lower limit = minimum variable cost per unit
Long-term lower limit = minimum average total cost
Short-term limit < Long-term limit always (fixed costs!)
Between the limits: Produce short-term, but not sustainable long-term

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Section Complete!

You’ve completed Section 08: Financial Mathematics. Review all formulas before the exam!

Homework Assignment

Tasks 08-03

Calculate variable cost per unit from various cost functions
Find short-term and long-term lower limit prices
Make production decisions based on price comparisons
Interpret contribution margins in business contexts

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These pricing decisions are frequently tested! Practice until you can solve them confidently.