[ Music ] >> To better understand the use of this fundamental equation, let's take a look at a numerical example and then discuss the implications of that example. So let's work through. Example one, the following information relates to a microchip manufactured by Kane Corporation. We have a variety of information available to us. First things first, how much are we selling it for? Well, we're told that the current selling price on a per unit basis is $7.55. Then we move on to direct costs. Direct labor on a per unit basis $1, and direct materials is $2. Both on a per unit basis. Variable manufacturing overhead, also on a per unit basis, is $1.35. And recall from Module One that those costs would include a variety of different sources, such as supplies and utilities. Other variable costs, outside of manufacturing, in this case dealing with selling costs, are provided on a per unit basis as well, and that's $1.20. So in all we have four different types of variable costs, direct labor, direct materials, variable manufacturing overhead and other variable costs. We also have fixed costs, some of them stem from the manufacturing process itself, and we're told that that amounts to $2.5 million per year. Other fixed costs allocated to the microchip amount to $1.5 million per year. So now let's take the equation and use it to apply this numerical information to calculate what Kane Company managers might be interested in. So recall the equation for the break-even point. We're told that the break-even point in units is equal to the total fixed costs in the numerator, divided by the contribution margin per unit, which is comprised of the selling price per unit, subtracting the variable costs per unit. If we were to take the information that's provided to us about Kane Company we can plug what we know into this equation to solve for that quantity that leads to the break-even point. In the numerator we would combine fixed costs. We were told that that was 2.5 million, and another source being 1.5 million. In the denominator we have our selling price and our variable costs. We were told that the selling price was $7.55 and the variable costs were direct labor, direct materials, other manufacturing overhead that was variable and other costs. Those were respectively, $1, $2, $1.35 and $1.20. Subtracting our variable cost from our selling price yields our contribution margin per unit. And so we basically have in our formula $4 million in total fixed costs and a contribution margin per unit that comes out to be $2. Dividing the $2 into 4 million yields 2 million units in order for Kane Company to break-even. We calculated the number of units that Kane had to produce and sell in order to break-even. So at the two million in production in sales mark, profits will be exactly zero. Fixed costs and variable costs will be completely offset by revenues. If Kane Company produces fewer than two million units, then the break-even point will not be reached, which means that profits for this year will be less than zero, will incur a loss. Production in sales that exceeds two million units will lead to profits because that level exceeds the break-even point. Now oftentimes a firm will ask more questions than just where we break even, but this is a great starting point, especially for firms as they enter into a new market. They might learn from this -- this calculation that they couldn't possibly produce two million units in a year; their capacity isn't sufficient and it would require a revision of the cost information. Or they might say, two million units per year is feasible so it's less risky to enter into this new market. So this is a nice starting point for CVP analysts to understand where losses start to become profits. So another reason why we went through a rather simple calculation is to look at some of the implications and the assumptions that underlie this calculation. First of all, it's assumed in our calculation that costs can be categorized as fixed or variable, or if they're mixed in some way, can be broken down appropriately. Recall in our equation that fixed costs are in the numerator and the variable costs are in the denominator. If there's any uncertainty about which costs are which, that might throw a wrench into our analysis. The output may not be as reliable because we had information about the cost structure wrong. Another assumption is that everything is linear. The selling price per unit was assumed to be $7.55, likewise we had estimates of what the variable costs per unit would be, and the fixed costs in total were going to be $4 million for the year. To the extent that any of this changes, or changes at different points in the production volume, our analysis is less reliable. Another assumption has to do with how we created the calculation. Recall in the steps of building that break-even point calculation that we assumed that the Q that was being multiplied by the selling price to yield revenues and the Q that was being multiplied by variable costs per unit to calculate total variable costs was the same Q. Another way of saying that is that we produce what we sell and we sell what we produce, there's no major change in inventory. To the extent that the amount of units or the number of units that is being applied to the -- via the sales price, and the number of units being applied to variable costs is different, our analysis becomes less reliable. There's a few more assumptions as well. One has to do with the production processes that the firm engages in. In essence we're assuming that the efficiency and the productivity of the production process remains constant. Now this assumption is always a bit of a stretch, because you can think of yourself during the day, how productive are you in the morning versus the productive in the afternoon and the evening. And of course, that waxes and wanes, but this assumption is necessary in order for us to rely on the analysis that CVP provides. And finally, two more assumptions that relate to a scenario that we haven't talked about yet, but since we're on the topic of assumptions we'll identify those. They have to do with multiproduct scenarios. And when we have multiple products, what we're going to assume about the situation is that the relative proportion of sales of our products, in other words the sales mix, remains constant over the relevant range that we're focused on. And since we produce what we sell and we sell what we produce, the production mix remains constant as well, and that does not change in response to sales volume changes. These assumptions, as we'll see in future examples, are crucial to the validity of our CVP analyses, and violations of these assumptions means that our outputs come into question.