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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
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CTC 475 Breakeven Analyses
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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
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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
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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
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Total Costs (TC) Total Costs (TC) at a unit of production = Fixed Costs (FC) + Variable Costs (VC) * # of Units Produced
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Fixed cost = $200 Variable Cost = $10 per unit
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Total Costs As currently defined total costs are linear with respect to units produced
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Can Decrease Costs by Lowering Fixed Costs ($200 to $150)
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Can Decrease Total Costs by Lowering Variable Cost ($10 to $8)
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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
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Total Revenues / Costs
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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
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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
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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
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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
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Linear Breakeven-Example
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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)
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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
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Price – Demand Relationship a-price at which demand=0 b-slope
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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
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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
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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)
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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
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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
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Maximize Revenue (Spreadsheet) TR = $800t-0.8t 2
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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
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Maximize Profit (Spreadsheet) NP=-0.8t 2 +400t-10,000
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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)
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Breakeven (Spreadsheet) t=26 & 474
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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
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Next lecture Estimates Accounting Principles
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