Session 08-03 - Cost Analysis & Pricing Decisions

Section 08: Financial Mathematics

Author

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

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 \(\overline{C}_v(x)\)
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 Feststellungsprüfung!

Cost Function Review

Components of Total Cost

Every business has two types of costs:

Cost Components

Fixed Costs (\(C_f\)): Costs that don’t change with production quantity

Rent, insurance, salaries, equipment depreciation

Variable Costs (\(C_v(x)\)): Costs that change with quantity produced

Raw materials, energy, direct labor, packaging

\[C(x) = C_f + C_v(x)\]

Total Cost Function Examples

Common cost function forms:

Linear: \(C(x) = 500 + 3x\): fixed cost 500, constant variable cost 3 per unit

Quadratic: \(C(x) = 400 + 10x + 0.02x^2\): variable costs increase with quantity

Cubic: \(C(x) = 1000 + 50x - 0.5x^2 + 0.01x^3\): the most realistic form, with economies and then diseconomies of scale

Visualizing Cost Functions

Two cost curves from 0 to 100 units, each with a dashed fixed-cost line and the variable cost shaded above it. Linear C(x) = 500 + 3x is a straight line from 500 to 800. Quadratic C(x) = 400 + 10x + 0.02x² bends upward from 400 to 1,600.

Average Cost

Average Total Cost

How much does each unit cost on average?

Average Cost (Durchschnittskosten)

\[\overline{C}(x) = \frac{C(x)}{x} = \frac{C_f + C_v(x)}{x} = \frac{C_f}{x} + \frac{C_v(x)}{x}\]

This can be written as: \[\overline{C}(x) = \underbrace{\frac{C_f}{x}}_{\text{Avg. Fixed Cost}} + \underbrace{\frac{C_v(x)}{x}}_{\text{Avg. Variable Cost}}\]

Variable Cost Per Unit

Variable Cost Per Unit (Stückkosten variabel)

\[\overline{C}_v(x) = \frac{C_v(x)}{x} = \frac{C(x) - C_f}{x}\]

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

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

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

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, it loses money on every unit!

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

Short-Term Limit

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.

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

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?

Minimum Variable Cost per Unit

Method: Use calculus to find the minimum of \(\overline{C}_v(x)\)

Step 1: Calculate \(\overline{C}_v(x) = \frac{C(x) - C_f}{x}\)

Step 2: Find \(\overline{C}_v'(x)\) and set equal to zero

Step 3: Solve for \(x\)

Step 4: Calculate \(\overline{C}_v(x)\) at this value

Alternatively, use: \(\overline{C}_v'(x) = \frac{C'(x) \cdot x - (C(x) - C_f)}{x^2}\)

Setting this to zero: \(C'(x) \cdot x = C(x) - C_f\)

Example: Short-Term Lower Limit

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

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

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

Example (continued)

Step 3: Find minimum \[\overline{C}_v'(x) = -0.2 + 0.004x = 0\] \[x = 50\]

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

Short-term lower limit price = 15 Euro per unit

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

Graphical Representation

Variable cost per unit and average total cost against quantity. The variable-cost curve is a U-shaped parabola with its minimum marked at (50, 15), the short-term lower limit price. The average total cost curve lies above it everywhere and falls steeply at small quantities.

Break

Long-Term Limit

Long-Term Lower Limit Price

Long-Term Lower Limit Price (Langfristige Preisuntergrenze)

The minimum price for sustainable long-term production.

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

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 \(\overline{C}(x) = \frac{C(x)}{x}\)

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

Step 3: Solve for \(x\)

Step 4: Calculate \(\overline{C}(x)\) at this value

Example: Long-Term Lower Limit

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

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

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

Example: Long-Term Lower Limit II

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 \(x^3 - 50x^2 - 200000 = 0\)):

Polynomial solver (MENU → A → 2, degree 3, Sessions 02-02 and 05-06): \(x \approx 80.7\)

Step 4: Calculate minimum average cost \[\overline{C}(80.7) = \frac{800}{80.7} + 20 - 0.2(80.7) + 0.002(80.7)^2 \approx 26.80\]

Long-term lower limit price = 26.80 Euro per unit

For sustainable production, the company must charge at least 26.80 Euro!

Short-Term vs Long-Term

The two unit-cost curves with their minima marked: variable cost per unit at (50, 15) and average total cost at about (80.7, 26.80). Horizontal bands show the pricing zones: below 15 never produce, between 15 and 26.80 short-term only, above 26.80 sustainable.

Interpretation

Pricing Decision Framework

Three Pricing Zones

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

When to Use Each Limit

Short-term lower limit price:

Factory/equipment 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: next month, its machines would stand still. A customer offers to buy 50 units at 16 Euro each.

With our cost function: short-term lower limit 15 Euro, long-term lower limit 26.80 Euro.

Analysis:

16 > 15 (short-term limit) \(\checkmark\)
16 < 26.80 (long-term limit)

Decision: Accept! The 50 units cost \(\overline{C}_v(50) = 15\) Euro each in variable costs, so each unit contributes 1 Euro toward fixed costs. Not all costs are covered, but this is better than nothing.

Contribution Margin

Contribution Margin (Deckungsbeitrag)

\[\text{Contribution per unit} = p - \overline{C}_v(x)\]

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

Special order (x = 50, the whole production): \(16 - \overline{C}_v(50) = 16 - 15 = 1\) Euro per unit, so the 50 units contribute 50 Euro toward fixed costs.

Use \(\overline{C}_v\) at the output of the order: for 100 units, \(\overline{C}_v(100) = 20 > 16\), and the same price would not even cover the variable costs.

Guided Practice

Practice Problems

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

Find the variable cost per unit function \(\overline{C}_v(x)\)
Find the short-term lower limit price (if \(\overline{C}_v\) only rises, use very small \(x\))
Find the long-term lower limit price

Practice Problems II

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

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

Exam-Style Problems

Typical Exam Problem Structure

FSP exam problems often combine these elements:

Given a cost function \(C(x)\)
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 \(\overline{C}_v(x)\)
“Langfristige Preisuntergrenze” = Long-term = min of \(\overline{C}(x)\)

Summary of Key Formulas

Essential Formulas for Cost Analysis

Concept	Formula
Total Cost	\(C(x) = C_f + C_v(x)\)
Variable Cost per Unit	\(\overline{C}_v(x) = \frac{C(x) - C_f}{x}\)
Average Total Cost	\(\overline{C}(x) = \frac{C(x)}{x}\)
Short-term Limit	\(\min\{\overline{C}_v(x)\}\)
Long-term Limit	\(\min\{\overline{C}(x)\}\)
Contribution Margin	\(p - \overline{C}_v(x)\)

Wrap-Up & Takeaways

Today’s Essential Concepts

Variable cost per unit \(\overline{C}_v(x)\) = 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

Section Complete!

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

Homework

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

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

Next Session Preview

Next Session: Mock Exam 09

Session 08-04 is exam only, no lecture

180 minutes working time plus 5 minutes reading time, 100 points
Covers the whole course so far: cost and profit analysis, curve sketching with integration, probability and binomial distribution
Allowed: non-programmable calculator, drawing instruments, monolingual dictionary
Not allowed: notes, formula sheets, any electronic devices
Counts toward the 60% of test points needed for exam admission

Preparing for Mock Exam 09

Complete Tasks 08-03, then prepare for Mock Exam 09 with FSP Training 08 (sidebar, Section 08 worksheets). Get a good night’s sleep and bring your calculator!