7-1

Break-even Example 1A firm produces radios with a fixed cost of $7,000 per

month and a variable cost of $5 per radio. If radios sell for $8 each:

1a) What is the break-even point?TR = TC so 8x = 7000 + 5xx = 7000/3 = 2,333.333 radios per month

1b) What output is needed to produce a profit of $2,000/month?Profit = 2000/month so TR - TC = 8x - (7000 + 5x) = 2000

x = 9000/3 = 3,000 radios per month

7-2

Break-even Example 1 - continued1c) What is the profit or loss if 500 radios are produced each

week?First, get monthly production:

50052/12 = 2,166.6667 radios per month

Then calculate profit or loss TR - TC = 82166.6667 - (7000 + 52166.6667)

= $-500 per month ($500 loss per month)

7-3

Break-even Example 2A firm produces radios with a fixed cost of $7,000 per month and

a variable cost of $5 per radio for the first 3,000 radios produced per month. For all radios produced each month after the first 3,000 the variable cost is $10 per radio (for added overtime and maintenance costs). If radios sell for $8 each:

2a) What are the break-even point(s)?Now TC has two parts depending on the level of production:For x 3000/month: TC = 7000 + 5xFor x > 3000/month: TC = 7000 + 5(3000) + 10(x-3000)

= -8000 + 10x For any x: TR = 8x

7-4

Break-even Example 2 - continuedFor x 3000/month: TC = 7000 + 5xFor x > 3000/month: TC = -8000 + 10xFor any x: TR = 8x

For x 3000/month: 7000 + 5x = 8x so x = 2,333.33/month This is < 3000/month, so it is a valid break-even point.

For x > 3000/month: -8000 + 10x = 8x so x = 4000/month This is > 3000/month, so it is also a valid break-even point.

7-5

Break-even Example 2

Total cost line

Total revenue line

1000

Break-even points

Volume (units/month)

Dol

lars

(Tho

usan

ds)

400030002000

8

24

32

16

40

7-6

Break-even Example 3A firm produces radios with a fixed cost of $7,000 per month and

a variable cost of $5 per radio for the first 2,000 radios produced per month. For all radios produced each month after the first 2,000 the variable cost is $10 per radio (for added overtime and maintenance costs). If radios sell for $8 each:

3a) What are the break-even point(s)?Again TC has two parts depending on the level of production:For x 2000/month: TC = 7000 + 5xFor x > 2000/month: TC = 7000 + 5(2000) + 10(x-2000)

= -3000 + 10xFor any x: TR = 8x

7-7

Break-even Example 3 - continuedFor x 2000/month: TC = 7000 + 5xFor x > 2000/month: TC = -3000 + 10xFor any x: TR = 8x

For x 2000/month: 7000 + 5x = 8x so x = 2,333.33/month This is not < 2000/month, so it is not a break-even point!!

For x > 2000/month: -3000 + 10x = 8x so x = 1500/month This is not > 2000/month, so it is not a break-even point!!

THERE ARE NO BREAK-EVEN POINTS!

7-8

Break-even Example 3

Total cost lineTotal revenue line

1000

Volume (units/month)

Dol

lars

(Tho

usan

ds)

400030002000

8

24

32

16

40

7-9

Other Break-even Possibilities

Total cost lineTotal revenue line

1000

Volume (units/month)

Dol

lars

(Tho

usan

ds)

400030002000

8

24

32

16

40

7-10

Crossover Chart

Total cost - Process CTotal cost - Process B

Total cost - Process A

Process A: Low volume, high varietyProcess B: Repetitive

Process C: High volume, low variety

Process CProcess BProcess A Lowest cost process

7-11

Crossover ExampleProcess A: FA = $5000/week VA = $10/unit

Process B: FB = $8000/week VB = $4/unit

Process C: FC = $10000/week VC = $3/unit

Over which range of output is each process best?

1. At x = 0 A is best (FA is smallest fixed cost).

2. As x gets larger, either B or C may become better than A:

B < A for x > 3000/6 or x > 500/week

C < A for x > 5000/7 or x > 714.28/week

so B is best for x > 500/week

3. Eventually, C will become better than B (VC< VB).

C < B for x > 2000/week

7-12

Crossover ExampleSummary:A is best for output of 0-500 units per week.B is best for output of 500-2000 units per week.C is best for output greater than 2000 units per week.

0 500 714 2000

A<B

A<C

B<CA<BA<CB<C

A<B<C

B<AC<AB<C

B<C<A

B<AA<CB<C

B<A<C

B<AC<AC<B

C<B<A

7-13

Crossover Chart

Fixed cost - Process A

Fixed cost - Process BFixed cost - Process C

Total cost - Process CTotal cost - Process B

Total

cost

- Pro

cess

A

Process A: low volume, high varietyProcess B: Repetitive

Process C: High volume, low variety

Process CProcess BProcess A Lowest cost process

7-14

Cost of Wrong Process Found Via Breakeven Analysis

Fixed cost

$

Variablecost

Fixed cost

$Variable

cost

Fixed cost

$Variable

cost

Low volume, highvariety process

Repetitive process High volume, lowvariety process

A B Volume

B1

B2B3

Total cost for lowvolume high variety

Total cost for repetitive processTotal cost for high volume,

low variety process

7-15

Time Value of Money - Net Present Value

¨ Future cash receipt of amount F is worth less than F today.F = Future value N years in the future.P = Present value today.i = Interest rate.

NN

i

FPiPF

)1()1(

7-16

Annuities¨ An annuity is a annual series of equal payments.

R = Amount received every year for N years.S = Present value today.S = RX

where X is from Table 7.5 (page 264).

Example: What is present value of $1,000,000 paid in 20 equal annual installments?

For i=6%/year, S = 5000011.47 = $573,500For i=14%/year, S = 500006.623 = $331,150

7-17

Limitations of Net Present Value

¨ Investments with the same NPV will differ:¨ Different lengths.

¨ Different salvage values.

¨ Different cash flows.

¨ Assumes we know future interest rates!

¨ Assumes payments are always made at the end of the period.

7-18

Limitations of Net Present Value¨ Investments with the same present value

may have significantly different project lives and different salvage values

¨ Investments with the same net present values may have different cash flows

¨ We assume that we know future interest rates - which we do not

¨ We assume that payments are always made at the end of the period - which is not always the case