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Understanding Breakeven Analysis for Profit Maximization

Learn how to recognize and solve breakeven analysis problems to maximize profit, minimize costs, and determine breakeven values in linear and nonlinear scenarios. Gain insights into fixed and variable costs, revenue optimization, and the time value of money.

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Understanding Breakeven Analysis for Profit Maximization

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  1. CTC 475 Review B/C ratios Use incremental; don’t rank

  2. CTC 475 Breakeven Analyses

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

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

  5. Fixed & Variable Costs • Fixed costs are expressed as one number • $200 • Variable costs are expressed as an amount 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 = $200Variable 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 variable cost • Increase revenue per unit

  15. Linear Breakeven Example

  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 ExampleDetermine 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 Relationshipp=a-b*D 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 • But p=a-bD (price and demand are related) • Therefore TR=(a-bD)(D) or • TR=aD-bD2

  22. Max. Revenue D high; p low High Sells Low revenue D low; p high Low Sells Low Revenue

  23. Maximizing Nonlinear Revenue • TR=aD-bD2 • Take derivative of TR w/ respect to D ; set derivative to zero and solve for D • Derivative=a-2bD=0 (will give zero slope) • D=a/2b • 2nd 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-0.8t2 • NP=TR-TC=-0.8t2+400t-10,000

  25. Maximize Revenue (Calculus) • TR = $800t-0.8t2 • 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-0.8t2

  27. Maximize Profit (Calculus) • NP=-0.8t2+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=-0.8t2+400t-10,000

  29. Breakeven (Algebra) • Set TC=TR and solve for t • -0.8t2+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 Revenues • Minimize Costs

  32. Time Value of Money • Most of this course is based on the fundamental concept that money has a time value • Must take this concept into account since projects have different cash flow patterns at different times

  33. Present Economy Problems • Time is not a significant factor

  34. Conditions: • No investment of capital • Long-term costs are the same for all alternatives • Alternatives have identical results

  35. Example-Present Economy • A metal part can be machined on an engine lathe (one at a time) or turret lathe (many at a time) Material costs are the same regardless of the machine used. • Parts are produced in batches according to the customer’s order • Based on the following cost data, what machine should be used for order sizes of 25, 100 and 500 units?

  36. Cost Data

  37. Turret Lathe Costs

  38. Engine Lathe Costs

  39. Summary • Turret lathe costs less per unit but has a high setup cost • Engine lathe costs more per unit but has no setup costs When do I use which machine?

  40. Comparison Costs Switchover occurs somewhere between 25 and 100 units (n=1 setup)

  41. Determine breakeven point-Math • 2.10x+48=3.45x • x=35 units (assumes N=1 setup)

  42. Determine breakeven-graphically

  43. Determine Breakeven Unit between 2 Machines with Math • Set Total Costs equal to each other between two machines: • Let Fixed Costs for Alternatives 1 and 2= FC1 & FC2 • Let Variable Costs for Alternatives 1 and 2=VC1 & VC2 • Set total costs equal to each other: FC1+VC1*D=FC2+VC2*D • Then D=(FC2-FC1)/(VC1-VC2) Note: embed this equation into the breakeven project

  44. Next lecture • Estimates • Accounting Principles

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