CTC 475 Review CTC 475 Review

Time Value of MoneyTime Value of Money Cash Flow Diagrams/TablesCash Flow Diagrams/Tables Cost Definitions:Cost Definitions:

Life-Cycle CostsLife-Cycle Costs Past and Sunk CostsPast and Sunk Costs Future & Opportunity costsFuture & Opportunity costs Direct and Indirect CostsDirect and Indirect Costs Average and Marginal CostsAverage and Marginal Costs Fixed and Variable CostsFixed and Variable Costs

CTC 475 CTC 475

Breakeven AnalysesBreakeven Analyses

ObjectivesObjectives

Know how to recognize and solve Know how to recognize and solve breakeven analysis problems:breakeven analysis problems:

Maximize profitMaximize profit Minimize costsMinimize costs Maximize revenuesMaximize revenues Determine breakeven valuesDetermine breakeven values Determine average costsDetermine average costs

Fixed and Variable Costs Fixed and Variable Costs

Fixed costs Fixed costs do notdo not vary in proportion to the vary in proportion to the quantity of output:quantity of output: InsuranceInsurance Building depreciationBuilding depreciation Some utilities Some utilities

Variable costs vary in proportion to quantity of Variable costs vary in proportion to quantity of outputoutput Direct LaborDirect Labor Direct MaterialDirect Material

Fixed & Variable CostsFixed & Variable Costs

Fixed costs are expressed as one numberFixed costs are expressed as one number $200 $200

Variable costs are expressed as an Variable costs are expressed as an amount per unitamount per unit $10 per unit$10 per unit

Total Costs (TC)Total Costs (TC)

Total Costs (TC) at a unit of production = Total Costs (TC) at a unit of production =

Fixed Costs (FC) + Fixed Costs (FC) +

Variable Costs (VC) * # of Units ProducedVariable Costs (VC) * # of Units Produced

Fixed cost = $200Fixed cost = $200Variable Cost = $10 per unitVariable Cost = $10 per unit

Costs vs. Units Produced

$0

$500

$1,000

$1,500

0 20 40 60 80 100

Units of Production

Co

sts

Fixed Variable Total

Total CostsTotal Costs

As currently defined total costs are linear As currently defined total costs are linear with respect to units producedwith respect to units produced

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

Costs vs. Units Produced

$0

$500

$1,000

$1,500

0 50 100

Units of Production

Co

sts

Original Total Costs Low er Fixed Cost

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

Costs vs. Units Produced

$0

$500

$1,000

$1,500

0 50 100

Units of Production

Co

sts

Original Total Costs

Low er Variable Cost

Total Revenue (Linear)Total Revenue (Linear)

Total Revenues = price (p) times number Total Revenues = price (p) times number of units sold (D)of units sold (D)

If I sell 100 units at $20 per unit then total If I sell 100 units at $20 per unit then total revenue = $2000revenue = $2000

Total Revenues / CostsTotal Revenues / Costs

Costs vs. Units Produced

$0

$500

$1,000

$1,500

$2,000

$2,500

0 20 40 60 80 100

Units of Production

Co

sts

Total Costs Total Revenues

BreakevenBreakeven

Breakeven occurs at the point where Breakeven occurs at the point where TR=TCTR=TC

If a company can sell more than the If a company can sell more than the breakeven point then the company makes breakeven point then the company makes a net profit (NP)a net profit (NP)

If a company sells less than the breakeven If a company sells less than the breakeven point then the company loses moneypoint then the company loses money

NP=TR-TCNP=TR-TC

Breakeven PointBreakeven Point

Ways to lower the Ways to lower the breakeven point:breakeven point:

Reduce fixed cost Reduce fixed cost Reduce variable Reduce variable

costcost Increase revenue Increase revenue

per unitper unit

Costs vs. Units Produced

$0

$500

$1,000

$1,500

$2,000

$2,500

0 20 40 60 80 100

Units of Production

Co

sts

Total Costs Total Revenues

Linear Breakeven ExampleLinear Breakeven Example

Turret Lathe: (Determine quantity needed to Turret Lathe: (Determine quantity needed to breakeven and net profit if 1000 units are sold)breakeven and net profit if 1000 units are sold)

One-Time Setup (FC)One-Time Setup (FC) $300$300

Material (VC)Material (VC) $2.50 per unit$2.50 per unit

Labor (VC)Labor (VC) $1.00 per unit$1.00 per unit

Selling PriceSelling Price $5.00 per unit$5.00 per unit

Linear BreakevenLinear Breakeven

Let D = # of Units that can be soldLet D = # of Units that can be sold TR = $5DTR = $5D TC = $300 + $3.50DTC = $300 + $3.50D

Set TR=TC and solve for D to find the Set TR=TC and solve for D to find the breakevenbreakeven

D=200 unitsD=200 units

Linear Breakeven-ExampleLinear Breakeven-Example

Costs vs. Units Produced

$0

$1,000

$2,000

$3,000

$4,000

$5,000

$6,000

0 200 400 600 800 1000

Units of Production

Co

sts

Total Costs Total Revenues

Linear Breakeven ExampleLinear Breakeven ExampleDetermine net profit (D=1000)Determine net profit (D=1000)

NP = TR-TCNP = TR-TC

TR=$5*1000 = $5000TR=$5*1000 = $5000

TC=$300+$3.5*1000 = $3800TC=$300+$3.5*1000 = $3800

NP=$1200 ($5000-$3800)NP=$1200 ($5000-$3800)

Nonlinear BreakevenNonlinear Breakeven

Usually there is a relationship between Usually there is a relationship between price (p) and number of units that can be price (p) and number of units that can be sold (D-for demand)sold (D-for demand)

If price is high demand is lowIf price is high demand is low If price is low demand is highIf price is low demand is high

Price – Demand Relationship Price – Demand Relationship

Price-Demand Relationship

$0.00

$0.50

$1.00

$1.50

$2.00

$2.50

0 200 400 600 800 1000 1200

Demand (gallons)

Pri

ce

($

)

a-price at which demand=0

b-slope

Price-Demand EquationPrice-Demand Equation

Price (p) = a – b *DPrice (p) = a – b *D Now let’s take a look at the TR equation:Now let’s take a look at the TR equation:

TR=pDTR=pD But p=a-bD (price and demand are related)But p=a-bD (price and demand are related) Therefore TR=(a-bD)(D) orTherefore TR=(a-bD)(D) or TR=aD-bDTR=aD-bD22

Total Revenue-Nonlinear

0

100

200

300

400

500

600

0 200 400 600 800 1000

Demand (gallons)

Tota

l Rev

enue

($)

D high; p lowSell many

Don’t make much revenue

D low; p highDon’t sell manyDon’t make much revenue

Max. Revenue

Maximizing Nonlinear RevenueMaximizing Nonlinear Revenue

TR=aD-bDTR=aD-bD22

Take derivative of TR w/ respect to D ; set Take derivative of TR w/ respect to D ; set derivative to zero and solve for Dderivative to zero and solve for D

Derivative=a-2bD=0 (will give zero slope)Derivative=a-2bD=0 (will give zero slope) D=a/2bD=a/2b

22ndnd derivative will tell you whether you have a derivative will tell you whether you have a max. (deriv. is neg) or min. (deriv. is pos) max. (deriv. is neg) or min. (deriv. is pos)

Breakeven Example - NonlinearBreakeven Example - Nonlinear

Given:Given: t is the number of tons sold per seasont is the number of tons sold per season Selling Price = $800-0.8t Selling Price = $800-0.8t TC=$10,000+$400tTC=$10,000+$400t Maximize revenue and profit; find breakeven pts.Maximize revenue and profit; find breakeven pts.

Calculations:Calculations: TR=Selling Price *t = $800t-0.8tTR=Selling Price *t = $800t-0.8t22

NP=TR-TC=-0.8tNP=TR-TC=-0.8t22+400t-10,000+400t-10,000

Maximize Revenue (Calculus)Maximize Revenue (Calculus) TR = $800t-0.8tTR = $800t-0.8t22

Set deriv = 0 and solve for t Set deriv = 0 and solve for t Deriv of TR w/ respect to t =800-1.6tDeriv of TR w/ respect to t =800-1.6t t=500 tonst=500 tons

Substitute t into TR equation to get TR=$200,000Substitute t into TR equation to get TR=$200,000 Substitute t into NP equation to get NP=$-10,000 Substitute t into NP equation to get NP=$-10,000 Lost money even though revenue was maximizedLost money even though revenue was maximized Better to maximize net profitBetter to maximize net profit

Maximize Revenue (Spreadsheet)Maximize Revenue (Spreadsheet) TR = $800t-0.8t TR = $800t-0.8t22

Total Revenue-Nonlinear

0

50,000

100,000

150,000

200,000

250,000

0 200 400 600 800 1000

Demand (tons)

To

tal

Rev

enu

e ($

)

Maximize Profit (Calculus)Maximize Profit (Calculus)

NP=-0.8tNP=-0.8t22+400t-10,000+400t-10,000 Set deriv = 0 and solve for t Set deriv = 0 and solve for t Deriv of NP w/ respect to t =-1.6t+400Deriv of NP w/ respect to t =-1.6t+400 t=250 tonst=250 tons

Substitute t into NP equation to get NP=$40,000Substitute t into NP equation to get NP=$40,000 Avg profit/ton=$40,000/250tons=$160 per tonAvg profit/ton=$40,000/250tons=$160 per ton

Maximize Profit (Spreadsheet)Maximize Profit (Spreadsheet) NP=-0.8t NP=-0.8t22+400t-10,000+400t-10,000

Net Profit-Nonlinear

-50,000

-30,000

-10,000

10,000

30,000

50,000

0 100 200 300 400 500

Demand (tons)

Net

Pro

fit

($)

Breakeven (Algebra)Breakeven (Algebra)

Set TC=TR and solve for tSet TC=TR and solve for t -0.8t-0.8t22+400t-10,000=0+400t-10,000=0 Must use quadratic equationMust use quadratic equation T=26 and 474 (if you sell within this range T=26 and 474 (if you sell within this range

you’ll make a net profit)you’ll make a net profit)

Breakeven (Spreadsheet)Breakeven (Spreadsheet)t=26 & 474 t=26 & 474

Breakeven Pts-Nonlinear

$0

$50,000

$100,000

$150,000

$200,000

$250,000

$300,000

0 200 400 600 800 1000

Demand (tons)

($

)

Total Revenue Total Costs

Tips to solve any type of breakeven Tips to solve any type of breakeven problem problem

TC=FC+VC (usually linear but could possibly be TC=FC+VC (usually linear but could possibly be nonlinear)nonlinear)

TR=p*D (may be linear or nonlinear)TR=p*D (may be linear or nonlinear) NP=TR-TCNP=TR-TC Breakeven pt(s) occur when TC=TRBreakeven pt(s) occur when TC=TR Maximize (or minimize) nonlinear equations by Maximize (or minimize) nonlinear equations by

finding derivative and setting equal to zerofinding derivative and setting equal to zero Maximize ProfitMaximize Profit Maximize RevenuesMaximize Revenues Minimize Costs Minimize Costs

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