1. CTC 475 Review  Time Value of Money  Cash Flow Diagrams/Tables  Cost Definitions: Life-Cycle Costs Life-Cycle Costs Past and Sunk Costs Past and Sunk Costs Future & Opportunity costs Future & Opportunity costs Direct and Indirect Costs Direct and Indirect Costs Average and Marginal Costs Average and Marginal Costs Fixed and Variable Costs Fixed and Variable Costs

2. CTC 475 Breakeven Analyses

3. Objectives  Know how to recognize and solve breakeven analysis problems: Maximize profit Maximize profit Minimize costs Minimize costs Maximize revenues Maximize revenues Determine breakeven values Determine breakeven values Determine average costs Determine average costs

4. Fixed and Variable Costs  Fixed costs do not vary in proportion to the quantity of output: Insurance Insurance Building depreciation Building depreciation Some utilities Some utilities  Variable costs vary in proportion to quantity of output Direct Labor Direct Labor Direct Material Direct Material

5. Fixed & Variable Costs  Fixed costs are expressed as one number $200 $200  Variable costs are expressed as an amount per unit $10 per unit $10 per unit

6. Total Costs (TC) Total Costs (TC) at a unit of production = Fixed Costs (FC) + Variable Costs (VC) * # of Units Produced

7. Fixed cost = $200 Variable Cost = $10 per unit

8. Total Costs  As currently defined total costs are linear with respect to units produced

9. Can Decrease Costs by Lowering Fixed Costs ($200 to $150)

10. Can Decrease Total Costs by Lowering Variable Cost ($10 to $8)

11. Total Revenue (Linear)  Total Revenues = price (p) times number of units sold (D)  If I sell 100 units at $20 per unit then total revenue = $2000

12. Total Revenues / Costs

13. Breakeven  Breakeven occurs at the point where TR=TC  If a company can sell more than the breakeven point then the company makes a net profit (NP)  If a company sells less than the breakeven point then the company loses money  NP=TR-TC

14. Breakeven Point  Ways to lower the breakeven point: Reduce fixed cost Reduce fixed cost Reduce variable cost Reduce variable cost Increase revenue per unit Increase revenue per unit

15. Linear Breakeven Example Turret Lathe: (Determine quantity needed to breakeven and net profit if 1000 units are sold) One-Time Setup (FC) $300 Material (VC) $2.50 per unit Labor (VC) $1.00 per unit Selling Price $5.00 per unit

16. Linear Breakeven  Let D = # of Units that can be sold  TR = $5D  TC = $300 + $3.50D  Set TR=TC and solve for D to find the breakeven  D=200 units

17. Linear Breakeven-Example

18. Linear Breakeven Example Determine net profit (D=1000)  NP = TR-TC  TR=$5*1000 = $5000  TC=$300+$3.5*1000 = $3800  NP=$1200 ($5000-$3800)

19. Nonlinear Breakeven  Usually there is a relationship between price (p) and number of units that can be sold (D-for demand)  If price is high demand is low  If price is low demand is high

20. Price – Demand Relationship a-price at which demand=0 b-slope

21. Price-Demand Equation  Price (p) = a – b *D  Now let’s take a look at the TR equation: TR=pD TR=pD But p=a-bD (price and demand are related) But p=a-bD (price and demand are related) Therefore TR=(a-bD)(D) or Therefore TR=(a-bD)(D) or TR=aD-bD 2 TR=aD-bD 2

22. D high; p low Sell many Don’t make much revenue D low; p high Don’t sell many Don’t make much revenue Max. Revenue

23. Maximizing Nonlinear Revenue TR=aD-bD 2 TR=aD-bD 2 Take derivative of TR w/ respect to D ; set derivative to zero and solve for D Take derivative of TR w/ respect to D ; set derivative to zero and solve for D Derivative=a-2bD=0 (will give zero slope) Derivative=a-2bD=0 (will give zero slope) D=a/2b D=a/2b 2 nd derivative will tell you whether you have a max. (deriv. is neg) or min. (deriv. is pos) 2 nd derivative will tell you whether you have a max. (deriv. is neg) or min. (deriv. is pos)

24. Breakeven Example - Nonlinear  Given:  t is the number of tons sold per season  Selling Price = $800-0.8t  TC=$10,000+$400t  Maximize revenue and profit; find breakeven pts.  Calculations:  TR=Selling Price *t = $800t-0.8t 2  NP=TR-TC=-0.8t 2 +400t-10,000

25. Maximize Revenue (Calculus)  TR = $800t-0.8t 2  Set deriv = 0 and solve for t  Deriv of TR w/ respect to t =800-1.6t  t=500 tons  Substitute t into TR equation to get TR=$200,000  Substitute t into NP equation to get NP=$-10,000  Lost money even though revenue was maximized  Better to maximize net profit

26. Maximize Revenue (Spreadsheet) TR = $800t-0.8t 2

27. Maximize Profit (Calculus)  NP=-0.8t 2 +400t-10,000  Set deriv = 0 and solve for t  Deriv of NP w/ respect to t =-1.6t+400  t=250 tons  Substitute t into NP equation to get NP=$40,000  Avg profit/ton=$40,000/250tons=$160 per ton

28. Maximize Profit (Spreadsheet) NP=-0.8t 2 +400t-10,000

29. Breakeven (Algebra)  Set TC=TR and solve for t  -0.8t 2 +400t-10,000=0  Must use quadratic equation  T=26 and 474 (if you sell within this range you’ll make a net profit)

30. Breakeven (Spreadsheet) t=26 & 474

31. Tips to solve any type of breakeven problem  TC=FC+VC (usually linear but could possibly be nonlinear)  TR=p*D (may be linear or nonlinear)  NP=TR-TC  Breakeven pt(s) occur when TC=TR  Maximize (or minimize) nonlinear equations by finding derivative and setting equal to zero Maximize Profit Maximize Profit Maximize Revenues Maximize Revenues Minimize Costs Minimize Costs

32. Next lecture  Estimates  Accounting Principles