MGS3100 Chapter 2: Slides 2d

Break-Even and Cross-Over Analyses

The Generalized Profit Model:

A decision-maker will break-even when profit is zero. We will develop the formula for the break-even point by setting the generalized profit model equal to zero, and then solve for the quantity (Q). For simplicity, we will assume that the quantity produced is equal to the quantity sold. This assumption will be relaxed in the chapter on decision theory.

Profit (π) = Revenue (R) - Cost (C)

Revenue (R) = Selling price per unit (SP) x Quantity sold (Qs)

Cost (C) = [Variable cost per unit (VC) x Quantity produced (Qp)] + Fixed Cost (FC)

For now, Let Qp = Qs

Therefore, π = R-C

π = (SP*Q)-[(VC*Q) + FC]

π = SP*Q - VC*Q - FC

π = (SP-VC)*Q - FC

If Contribution Margin (CM) = SP-VC, then…

π = CM*Q - FC, and

Q = (FC + π)/CM

From this formula, we can determine the quantity to produce and sell that will yield a profit of π dollars. For example:

If fixed cost is $150,000 per year, selling price per unit (SP) is $400, and variable cost per unit (VC) is $250, what quantity (Q) will produce a profit of $300,000?

CM = SP - VC = 400 - 250 = 150

Q = (FC + π)/CM = (150,000 + 300,000)/150 = 450,000/150 = 3000 units

Breakeven Point:

By setting π = 0 and solving for Q, we find the break-even quantity, as follows:

0 = CM*QBE - FC

FC = CM*QBE

FC/CM = QBE

QBE = FC/CM

For example, if:FC = 150,000VC = 250SP = 400

QBE = 150,000/150 = 1000 units

Pictorially:

ΠBreak-Even Point

01000 Q

-150,000

} CM = 150

Cross-Over Point (Indifference Point):The cross-over (or “indifference”) point is found when we are indifferent between two plans. In other words, this is the value of Q when profit is the same for each of two plans. To find the cross-over point for Plan A and B:

πA = CMA*QA - FCA

Set πA equal to πB and solve for value of Q πB = CMB*QB - FCB

Let’s call this “cross-over point” (between Plans A and B) Q “AtoB” (or, in general, “Qco”). Therefore, setting the

two equations equal to each other and solving for Q:

QAtoB = (FCA - FCB)/(CMA - CMB)

To illustrate the cross-over point, let’s look at three strategies or plans:

Plan A Plan B Plan CFC 150,000 450,000 2,850,000VC 250 150 100SP 400 400 400

Breakeven Points for each plan are:

Plan A Plan B Plan CQBE = 150,000/(400-250) 450,000/(400-150) 2,850,000/(400-100)

= 1000 units = 1800 units = 9500 units

And, by definition, the profit at each break-even point is zero.

Crossover Points:

A to B B to CQCO (150,000-450,000)/(150-250) (450,000-2,850,000)/(250-300)

= 3000 units = 48,000 units

And since π = CM*Q - FC, the profit at each cross-over point is:

} }150 250 } 300

πB = CMB*Q - FCB

= 250(48,000) - 450,000

= $11,550,000, or

πC = 300(48000) - 2,850,000

= $11,550,000

πA = CMA*Q - FCA

= 150(3000) - 150,000

= $300,000, or

πB = 250(3000) - 450,000

= $300,000

To interpret the answers, we are “indifferent” between Plan A and Plan B when Q = 3000 units, and either Plan A or B would yield a profit of $300,000 when Q = 3000 units. We are also “indifferent” between Plan B and Plan C when Q = 48000 units, and either Plan B or C would yield a profit of $11,550,000 when Q = 48000 units. Pictorially:

KEY:ABC

Π NOT TO SCALE

3,000 48,000

Crossover Points

Below 1000 units, none of the strategies would break-even. And remember that the generalized profit model above can be used to find the amount of profit for a plan at any value of Q. Therefore, depending on the number of units produced and sold (Q), the best plan to pursue would be as follows:

Units (Q) Decision0-999 Don’t produce

1000-2999 Plan A3000 Plan A or Plan B

3001-47,999 Plan B48,000 Plan B or Plan C

>48,000 Plan C

Q