Session 08-03 - Cost Analysis & Pricing Decisions

Section 08: Financial Mathematics

Dr. Nikolai Heinrichs & Dr. Tobias Vlćek

Entry Quiz

Quick Review from Session 08-02

Test your understanding of Annuities

  1. Calculate the future value of saving 250/month for 15 years at 4% annual interest.

  2. Find the monthly payment for a 30,000 loan at 5.4% for 4 years.

  3. After 12 payments on a 15,000 loan at 6% annual for 3 years, what is the outstanding balance?

  4. 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 \(C(x) = 1200 + 30x + 0.1x^2\)

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

Practice Problems II

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

  1. What is the short-term lower limit price?
  2. What is the average total cost at a production of 100 units?
  3. 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:

  1. Given a cost function \(C(x)\)
  2. Find the break-even point(s) given price \(p\)
  3. Find the profit-maximizing quantity
  4. Find the short-term/long-term lower limit price
  5. 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!