The Three Choice Problems

# Three basic questions

## Break-even question

When we perform a sensitivity analysis of a project, we are asking how serious the effect of lower revenues or higher costs will be on the project's profitability. Managers sometimes prefer to ask instead how much sales can fall below forecasts before the project begins to lose money. This type of analysis is known as break-even analysis. In other words, break-even analysis is a technique for studying the effect of variations in output on a firm's NPW (or other measurements). We will present an approach to break-even analysis based on the project's cash flows.

Transcript of Break Even Problem Video

### A slightly advanced example with time value of money

To illustrate the procedure of break-even analysis based on NPW, we use a generalized cash flow approach. We compute the PW of cash inflows as a function of an unknown variable, perhaps annual sales. For example,

PW of cash inflows = f(x)1

Next, we compute the PW of cash outflows as a function of x:

PW of cash outflows = f(x)2

NPW is the difference between these two numbers. Accordingly, we look for the break-even value of x that makes

f(x)1 = f(x)2

An important point to make is that the domain of x >= 0. If the calculated break-even value is negative, then the break-even point is 0. (Since one cannot create a negative quantity of product, then the best one can do in this situation is create none at all.)

An interesting example where a company produced a product that sold for less than it's cost is Sony with the PS3: the units were originally produced at about \$900 per unit, but sold for \$499 when they first hit the market; this made the PS3 one of the best Blue-Ray players for under a \$1000 to purchase at the time. The reason for selling less than cost was the expected return in revenue from the games purchased: DVDs are cheap to replicate in mass quantities, but the games easily sold for over \$50 per DVD (which probably cost them less than \$1 per DVD to replicate).

Note that this break-even value is similar to that used to calculate the internal rate of return when we want to find the interest rate that makes the NPW equal zero. The break-even value is also used to calculate many other similar "cutoff values" at which a choice changes.1

• Finding the cheapest technology, given a known quantity

## Optimal Pace/Rate Question

• Optimal pace/rate is the quantity that maximized net benefits
• If incremental benefits are greater than or equal to incremental costs, "Do it."
• Incremental benefits (IB) should be ordered from largest to smallest. IB function is not increasing
• Incremental costs (IC) should be ordered from smallest to largest. IC function is non-decreasing.

### Optimal Pace/Rate Example Problem:

Jamie owns a small business where he completes a wide array of tasks for his clients. He has five workers who he hires (as needed) to complete these tasks. One Monday, when he comes into the office he has a list of tasks requested from his clients. Each task takes one man and one full day. Which tasks should he agree to do and which men should he employ to maximize his benefits?

 Workers Wages Tasks Pay Larry \$9 Shine Shoes \$8 Curly \$5 Paint a Fence \$12 Moe \$7 Register Voters \$4 Shemp \$1 Wait Tables \$5 Joe \$2 Babysit \$14

First order the workers (incremental costs) from smallest to largest and the tasks (incremental benefits) from largest to smallest.

 1 Shemp \$1 Babysit \$14 2 Joe \$2 Paint a Fence \$12 3 Curly \$5 Shine Shoes \$8 4 Moe \$7 Wait Tables \$5 5 Larry \$9 Register Voters \$4

Match each worker with the associated task until their cost is greater than their benefit

 Combination Profit Shemp + Babysitting \$14 - \$1 \$13 Joe + Painting a Fence \$12 - \$2 \$10 Curly + Shining Shoes \$8 - \$5 \$3 Moe + Wait Tables \$5 - \$7 -\$2

Thus we would hire Shemp, Joe, and Curly and we would take the tasks of babysitting; fence painting; and shoe shining. This choice would net us the maximum profit of \$26.

Another Optimal Rate/Pace Example
Brian's Delivery Service

Yet Another Example
Midterm optimal rate Question

Quiz 2/18/2010

Consider the following three production technologies.

Technology A Has a fixed cost of \$320.00.
Technology B Has a an average variable cost of \$16.00.
Technology C Has a fixed cost of \$40.00 and an average variable cost of \$14.00.

In all cases assume that you can sell your output for \$20.00.

Which technology would you use when your output level is 20?

(a) A
(b) B
(c) C
(d) Doesn't matter

First we want to set up a chart showing the total costs (fixed costs + average variable cost per unit) associated with each technology and the total benefits gained when producing 20 units at a cost of \$20/unit.

Selling 20 units @ \$20 = \$400 Net Income. Subtract Total Costs from Net Income to receive Total Benefits.

Total Cost Total Benefits
Tech A \$320 \$400 - \$320 = \$80
Tech B \$16(20) = \$320 \$400 - \$320 = \$80
Tech C 40 +14(20) = \$320 \$400 - \$320 = \$80

We gain the same most amount of benefits by using Technology A, B & C: \$80 so we choose (d)

Which choice should we choose if we produce one unit more?

You could do the whole process over again but I usually just conceptualize at this point.

If you have reached equal benefits with all options (all options are equal) then look for the option with ONLY fixed costs. That one doesn't cost more if you produce one more unit. So we choose Technology A

Which choice should we choose if we produce one unit less?

We rule out that option A cost too much because it was the right choice for the last question. Thus, Option B and C are to be compared.

Since Fixed costs are not changing you might think all we need to check at this point is the variable cost per unit

Tech B is \$16 per unit
Tech C is \$14 per unit

Since Tech C is cheaper to produce than Tech B by \$2 (ie \$16-\$14=\$2) But if you check it by plugging it into the equation you will see that option B is a better choice because option C's \$40 fixed costs make it slightly more expensive.

I point all of this out to remind us that our intuition, though useful for many things, can cause us to assume incorrectly and sometimes it is wiser to just work out the problem.

Total Cost Total Benefits
Tech B \$16(19) = \$304 (cost) and \$19*20 =\$380 (benefit) , so Total profit \$380 - \$304 = \$76
Tech C \$40 +\$14(19) = \$306(cost) and \$19*20 =\$380 (benefit) , so Total profit \$380 - \$306 = \$74

Thus we choose the option with the highest Net Benefits: option B

page revision: 2, last edited: 17 Aug 2016 18:01