Two Perspectives on Profit and Loss

You have one profit number and two different questions to answer with it - how did we do, and what happens if we grow? Which way you cut your costs depends on which question you’re asking.

The Brief

Why this matters. Profit is just revenue minus cost - but how you analyze and communicate that profit depends on your perspective. Report last quarter to a stakeholder and you need one way of cutting costs; decide whether to buy the machine or double production and you need another. Managers who only know one lens keep answering the wrong question with the right number.

Key terms. income statement (P&L), cost of goods sold, gross profit, SG&A, EBITDA, depreciation and amortization, EBIT, net profit, contribution margin, break-even quantity, operating leverage, degree of operating leverage, margin of safety.

The Saturday lemonade stand

Two friends set up lemonade stands on the same Saturday. Ava rents a fancy cart for $150 for the day. Her variable cost is $0.50 a cup, and because her stand looks nicer, she can charge $1.50. Ben sets up a fold-up table - no fixed costs at all, the same $0.50 a cup in variable costs, but with the quick-and-dirty setup he can only charge $1.00.

Which one would you rather be? Hold that question - by the end of this chapter you can answer it precisely, and the answer depends on something neither friend controls.

4.1 One equation, two lenses

The profit equation never changes - what changes is how you classify the costs, and which classification you need depends on the decision in front of you.

Profit is what we have left. The customers give us money in for what we do - that’s the revenue. Costs are what we have to expend in order to capture those revenues. The difference, the surplus, is profit, which is a proxy for value creation. Revenue decomposes into price times quantity. Costs are where it gets interesting - there is more than one way to cut them.

One of the important things about understanding this chapter is to really understand: are you making money? The report card is understanding the language of business - revenues minus costs in a coherent way, across what kind of costs we are incurring and what kind of profit we have after each activity. For that, you need the income statement view. However, sometimes you need to make decisions about extent - whether or not we’re going to make more, how much we’re going to make, pricing decisions. Should I buy this machine? Should I invest in this capability? What kind of volume will I achieve from that? That requires cutting costs in a different way. That is the cost behavior view, where we break things down into fixed cost and variable cost.

You need both lenses to make smart decisions - and, very important, you need to know when one is the better perspective versus the other, depending on the decision you’re going to make.

4.2 The income statement: where did the money go?

The P&L subtracts costs in a deliberate, stepwise order - from the costs inside the product to the most peripheral - so that each subtotal answers a question someone cares about.

When a company wants to report its performance - to managers, to owners, to lenders - it turns to the income statement, also called the statement of financial results or simply the P&L. Here is a simplified one:

We start at the top line, which is revenue, and then we’re going to subtract cost - but we’re going to do it in a very stepwise, specific manner.

Account	Period 1
Revenue (Sales)	$100,000
Cost of Goods Sold (COGS)	$60,000
Gross Profit	$40,000
Selling Expenses	$10,000
General & Admin Expenses	$8,000
EBITDA	$22,000
Depreciation & Amortization	$2,000
EBIT (Operating Income)	$20,000
Interest Expense	$1,000
EBT	$19,000
Taxes	$4,500
Net Profit	$14,500

The first thing we subtract off is cost of goods sold. That is all the costs that go into the product. If you’re making lemonade: lemon, sugar, water, the paper cup - that’s all the stuff that’s in the product that you had to procure or buy in order to provide it. The definition is in the name: these are the costs where you can directly trace them into the good that you’re selling. Subtract them and what you get is gross profit. That’s a profit number, but it doesn’t have everything in it, so we call it gross - a rough profit number. The question we’re answering here is: is the product profitable?

The thing is, you are running a business, not just making a product - you’re often selling the thing too. How do you find customers? How do you talk to customers? How do you manage the business? How do you keep track of things? There are going to be expenses that come with running the business that are just not in the product itself: paying your salespeople, paying accounting clerks, paying general managers, the expenses of having a store to sell it in. These are the selling, general, and administrative costs - SG&A. Subtract those off and what you have left is another subtotal, EBITDA: earnings before interest, taxes, depreciation, and amortization. Again, it’s right in the name - it’s earnings, but before we account for some stuff.

Now, depreciation and amortization is the cost of when you buy something really expensive. We assume that every single year it degrades a little bit, and we take that as an expense - the expense of using this long-lived equipment. Depreciation is for equipment; amortization is for other things with a long life, like intangible assets. It’s not actually cash that we pay, but because we’ve used this stuff up, we allocate an expense to it. Subtract that and we go from EBITDA to EBIT - earnings before interest and taxes. Subtract off interest expense, and the next line is earnings before taxes, because businesses get taxed on their profits, not on their revenue. Take the tax off and that is net profit.

You notice it sits on the bottom - it’s what accountants like to call the bottom line. The whole vernacular of “the bottom line,” the final thing at the end, comes from accounting language. Net profit is what the business has after we subtract all the different things it takes to run the business, including taxes. If this were a single owner, it’s what the owner can either put in their pocket or reinvest in the business.

We’re communicating how this business performed to stakeholders - the managers who want to know “is this business running well?”, the owner, or, if it’s a publicly traded company, the investors. This cost breakdown, along with reporting the profits, is the primary way we communicate that. It answers: what happened, and where did the money go?

Bottom line: the income statement is the language of performance - it categorizes costs by purpose so you can explain the past to people who need to know.

4.3 The cost behavior lens: what a single sale earns

Reclassify the same costs by how they respond to volume, and you can answer a different question - what happens to profit if we grow?

Now there’s another way of thinking about profits. We’re only changing how we classify costs - it’s still revenue minus costs. But instead of classifying costs by product cost, then selling cost, then depreciation and interest, we classify them into just two areas, by how they change when the business grows or shrinks: variable costs, which grow as we sell more, and fixed costs, which don’t. The cost chapter (Chapter 3) built that distinction carefully; here we put it to work.

The core concept is the contribution margin: price minus the variable cost. You sell one unit - say you sell a glass of lemonade - you get some revenue, but you have to pay for the cup, the lemons, the sugar, the water, and then whatever’s left is kind of a profit number too. It’s what we have in our hands after we pay for those costs. But then we have a little bit more - we have to pay for our fixed costs, and then whatever is left is profit. So whatever we have left goes to one or two different sources: it has to pay down our fixed cost that we incur up front, and whatever is left after that becomes our profit.

That leads to the break-even quantity. Think about this: when you just first sell, the problem is you’ve incurred all this fixed cost. You’re kind of in the hole - and then you sell just one unit, and you’re still in the hole because you haven’t recovered all your fixed cost yet. Break-even is how many units do I have to sell to kind of dig myself out of this hole and just be back to zero? Every other unit we sell after that, that’s profit.

Worked example: who wins the Saturday, Ava or Ben? Go back to the anchor case: Ava has $150 of fixed costs, charges $1.50, and pays $0.50 a cup. Ben has no fixed costs, charges $1.00, and pays the same $0.50 a cup.

\[ CM_{Ava} = \$1.50 - \$0.50 = \$1.00 \qquad CM_{Ben} = \$1.00 - \$0.50 = \$0.50 \]

Ava contributes twice as much per cup - but her contributions have a $150 hole to fill first, and Ben’s go straight into his pocket.

Below 150 cups, Ava is losing money on the day. Ben breaks even on his very first cup - with no fixed costs, he was never in the hole.

Step 3 - profit at a given volume. Profit is contribution times quantity, minus the fixed cost:

At 50 cups: Ava earns $(1.00)(50) - 150 = -\$100$ while Ben earns $(0.50)(50) = \$25$. At 500 cups: Ava earns $\$350$ to Ben’s $\$250$.

Step 4 - the crossover. Set the two profit equations equal and the volume where they tie is 300 cups. Below it, Ben’s flexibility wins; above it, Ava’s margin wins.

Bottom line: contribution margin tells you what each sale earns; break-even tells you how many sales dig you out of the fixed-cost hole; and which cost structure wins depends entirely on volume.

4.4 P&L lines are not behavior categories

The income statement’s categories have meaning, but fixed and variable costs are mixed inside every one of them - you have to take each line and ask how it behaves.

Now, do these things map onto each other nice and neat? Well, not really. If we look at the income statement lines, they don’t quite equal cost behavior.

Cost of goods sold - a lot of it’s variable, but it might include some fixed component. Go back to the lemonade. When we talk about selling lemonade, what’s directly in this product? What’s in that cup is water, lemon, sugar, and the paper cup itself. These are variable components - we’re buying stuff to make stuff. What else is in there is, let’s say, the labor of the person who’s mixing up the lemonade. Now this person we might hire on a fixed basis, for a fixed number of hours every month or every year. We’re going to hire this person whether we have lemonade to make or not, because we made an employment commitment with this person. If you think about it, that person’s hours of labor are also in that lemonade cup. So even though cost of goods sold - a lot of it, especially the material stuff - is often variable, because we can just buy more stuff or less stuff depending on how much we sell, there can be some stuff in cost of goods sold that is fixed.

If we go below gross profit into selling, general, and administrative costs, there is a mix of fixed and variable things. Some things might be fixed, like the rent that we pay on our store or salaries that we pay people, but there could be some variable things too, if salespeople are paid on a commission - the more they sell, the more they get. Shipping costs: if we make stuff that we ship through UPS or FedEx, we only pay them when we ship things, so in some sense they are variable. Depreciation charges don’t change, because they just depend on the cost of the asset, irrespective of the volume. Even interest expense can vary if we’re drawing on a revolving credit line tied to sales.

That’s to say, as you go down the profit and loss statement, these categories have some meaning, but some of what’s inside is fixed and some is variable. You have to be thoughtful about reclassifying these costs. A smart manager doesn’t stop at the P&L. You need to take each profit and loss line and think about how this cost behaves as the business changes.

Bottom line: the P&L tells you where the money went by category; only reclassifying each line by behavior tells you what will happen when volume changes.

4.5 Operating leverage: profit moves most when it is thinnest

Fixed costs magnify the effect of volume on profit - enormous gains or losses near break-even, calmer ones far from it.

Here’s the intuition. When you have high fixed costs, it’s like digging yourself into a hole from the very beginning, and you have to find some way to dig yourself out by selling. The upside is that if you sell a lot, you can make a lot of money - once you’ve covered your fixed costs, everything is pure profit. If you don’t sell enough, you really risk losses. For low fixed costs, the good news is you’re flexible: more business means more cost, less business means less cost. The issue is that with high variable costs, your margins might be small - growth is good, but the profit that comes with growth is slower.

Why call it leverage? It’s kind of like when small moves in one place have big-size outcomes in a different place - that’s when we think about a lever. With a small move you can pry loose something really heavy. “Increase effort by 20% and get a 20% increase in revenues” - that’s expected, understood, intuitive. Operating leverage is when a couple of small moves in volume produce outsized swings in profit, and that disproportion is exactly what makes it worth studying.

And it swings both ways. I once heard from a media executive that operating leverage is awesome and feels really good going up - you’ve invested a lot upfront, you see volumes soar, and profits just kick in. When you’re growing, it makes sense to hire the specialist who writes the reports, the one person who plans events for everybody. With every step you’re increasing efficiencies, and everything you do seems magical: more, better, at lower cost. The problem is when volume starts going down. You’ve got all these people whose capacity is left fallow - they contribute a lot when there are lots of customers, but they don’t bring customers in themselves. Now you carry the full cost burden of resources you can’t easily undo, and profitability slips fast. It’s great to accumulate leverage as you grow, and it feels just as bad on the way down. That’s the risk you’re always balancing.

So why get into the math, if we understand this conceptually? Because the specificity is where things really matter. It’s not enough to know “high fixed costs could be great when there’s high volume” - you run the math and maybe it’s not so great, and you’re taking on a lot of risk for something that won’t provide much benefit. Or the opposite: maybe the risk isn’t bad at all, because the volume you’d need is so much less than what you’re expecting that it’s a no-brainer. I was working with a musician once who did very small concerts, and we thought about getting some operating leverage - bigger audiences, larger venues. A larger hall is another big fixed cost, unprofitable unless you fill enough of the seats to make it worth it. When we crunched the math, it turned out she’d have to fill a really large percentage of a hall to make it barely worth it - very different from what we could have seen coming in. It took us five minutes to figure that out, rather than learning it the hard way with a down payment on a hall. The only way to really figure this out is to put the numbers down and check them against your intuition.

The static formula. Define the terms: $Q$ is quantity sold, $P$ is price per unit, $VC$ is variable cost per unit, $FC$ is fixed costs, so operating profit is $\pi = (P - VC)\,Q - FC$. The degree of operating leverage (DOL) is our measure of how much of this disproportionate effect is in play. In words, you’re taking the total contribution margin over the operating profit:

\[ \text{DOL} \;=\; \frac{\underbrace{(P - VC)\,Q}_{\text{total contribution margin}}}{\underbrace{(P - VC)\,Q - FC}_{\text{operating profit}}} \]

Every time you look at a formula, go slow over it, because there are insights to have. There’s something that’s the same in the numerator and denominator - the contribution margin, $(P-VC)\,Q$, in both the top and the bottom - and the only thing different in the bottom is that we subtract out the fixed cost. So first: if we had no fixed cost, everything 100% variable, this would just be one. It’s linear - increase the size of the business and costs, revenue, and profit all grow at the same percentage. No leverage. But when fixed cost is a really big portion of the cost structure, something interesting happens: the denominator shrinks, and the ratio grows. If you think about this math, it looks most dramatic right when you cross the break-even point - right when you’ve finally paid off your fixed cost, every single unit you sell increases profits dramatically as a percentage of what you had before, and then it levels off.

So if someone hands you a DOL of four, what have they told you? The only way you get numbers like that is a pretty sizable fixed cost - a denominator that’s small against a numerator that’s big. It means you’re above break-even, but right above that zone, where every piece of volume makes a big difference - so you have to be very cognizant of preserving and protecting that volume, because when volume constricts, it’s going to be really painful.

Watch it work in a theater. A movie theater is the canonical example of lots of fixed costs and very incremental variable costs. Say you have 100 seats, and you have to fill 60 of them just to pay for the rent - that’s the break-even. You’re making a loss the first minute you have this theater with no tickets sold, and every ticket in the march up to seat 60 goes to paying out the fixed cost. At seat 60, you’ve finally paid for all of it. Now the 61st seat takes you from zero profit to the full contribution margin - it means a lot, because anything is better than nothing. The 62nd seat doubles your profit. The 63rd raises it 50% from what you just had; the 64th, about a third; and it gets lower and lower. Still great to be increasing profits - every contribution matters - but the more you accumulate, the less the incremental unit hits as hard. By seat 95, you’ve already got so many wins that every extra ticket adds just a bit more on top. That’s the shape of leverage: a big difference-maker right around break-even, fading as you max out.

[Exhibit omitted from this web edition pending a rights-cleared version.]

Degree of operating leverage for the 100-seat theater. DOL explodes just past the 60-seat break-even and decays toward 1 as the house fills.

The elasticity version. Students ask why we have two different formulas - and the second one is more complicated, with concepts of calculus in it. I try not to add complexity needlessly; we’re not doing calculus just to do calculus. There’s something in this one we didn’t see in the first. The first formula showed how fixed cost plays a role. This one is the true expression of what leverage and sensitivity mean - the percentage change in profit over the percentage change in quantity:

\[ \text{DOL} \;=\; \frac{\%\Delta \pi}{\%\Delta Q} \;=\; \frac{d\pi / \pi}{dQ / Q} \;=\; \frac{d\pi}{dQ} \cdot \frac{Q}{\pi} \]

Look at the piece on the right. The slope, $d\pi/dQ$, is constant - every single unit we sell, we get the contribution margin, price minus variable cost, in dollars. So why does leverage change? Because the elasticity depends on where we are. Right at break-even you’re making zero, and one more unit is - as a percentage - infinitely more profit than before. The more you get beyond break-even, the larger the base on which you’re increasing, while the profit from each unit stays constant. As a percentage of what we have, it starts shrinking. That’s why the constant slope gets multiplied by $Q/\pi$ - by where we are on the curve. DOL is a function not just of what the slope is, but of where we are on that volume trade-off.

Two versions of the truth, reconciled. I never liked having two versions of the truth - if we’re talking about the same thing, it’s important to demonstrate it is the same thing, and math allows us to do that. Take the profit function:

Ask how profit changes when $Q$ changes - differentiate with respect to $Q$:

The $Q$ drops out, the fixed costs drop out, and what’s left is just the contribution margin per unit. Substitute that into the elasticity definition:

\[ \text{DOL} \;=\; (P - VC)\cdot\frac{Q}{(P - VC)\,Q - FC} \;=\; \frac{(P - VC)\,Q}{(P - VC)\,Q - FC} \]

Exactly what we had in the static formula. Two different cuts of the same thing - and each cut tells us a different story. The static version shows you the role of fixed costs; the elasticity version shows you that where you sit relative to break-even is what makes the number move.

What people get wrong. Students - and frankly, managers - often get operating leverage wrong in one of two directions. The optimists think it’s simply good: every additional unit feels like it’s going to be a lot of profit. That’s not quite right, because the magnification only really matters around the break-even point. The overly risk-averse get it wrong the other way: leverage means fixed costs, fixed costs mean risk, so minimize all fixed costs. That can get you in trouble too - if you’re not taking advantage of these resources, you pay a lot more in variable costs, your margins are smaller when you grow, and it’s a real drag on growth. It’s not about good or bad. There has to be a balance of risk, and you need to know the math and the context you’re in, crunch the numbers, and make a judgment.

The judgment tool is the margin of safety: compare the volume you actually need to break even against the volume you expect. We don’t live in a precise world, and we’re not targeting break-even. If your expected volume is way over break-even, you have a large margin of safety - you could be wrong by a lot and still make money, so you’re more comfortable making the move. If your prediction is really close to break-even, then being just a little bit wrong means much less profit than you think, maybe losses - a very slim margin of safety. That doesn’t mean don’t do it; it means you need to be really okay with the risk, or much more confident in your estimates. So: you want operating leverage when you’re very confident you’ll achieve volume comfortably above break-even - you allow yourself to be more efficient and make more money. You don’t want a lot of it when volume looks low or volatile - low leverage keeps you flexible enough to make profits whether volumes are high or low, but that flexibility comes at a cost, because your contribution margins are likely smaller and the upside of high volume is limited by the choices you made.

Bottom line: leverage is an amplifier, not a verdict - the math tells you how hard volume will swing your profit, and the margin of safety tells you whether you can live with the swing.

Bringing it back

So which one would you rather be? It depends on the volume. If you’re going to sell a lot of cups, you’d rather be Ava - even with the fixed cost, you make more money in the long run. If you’re not going to sell very many, you’d rather be Ben, who doesn’t carry the burden of the fixed costs and, even with a lower margin, comes out ahead. The crossover sits at 300 cups, and everything hinges on which side of it Saturday lands on. That is how a manager uses both lenses together. Start with the P&L - we always want to know: where did the money go? What happened? Then you reclassify costs into behavior: what are the things that vary when we sell more, and what are the things that just stay fixed? Do this as best as possible. Then break down the business into one sale - how much are we making on each sale? That’s the price minus the variable cost; that is the contribution every time we sell one of these things. From here we can assess leverage: how much of our costs are fixed, what is our break-even, what is our margin of safety. And then we decide - from this insight, we connect it with our P&L story and look forward to say what it is we should do, because the stakeholders you’re persuading speak the language of the income statement.

Check your understanding

4.5.1 Concept checks

4.5.2 Apply it

4-1 The Saturday Crossover. [LO2] Using the anchor case numbers (Ava: FC $150, P $1.50, VC $0.50; Ben: FC $0, P $1.00, VC $0.50): (a) verify the 300-cup crossover by setting the two profit equations equal; (b) compute each friend’s profit at 100, 300, and 600 cups; (c) state in one sentence the decision rule you would give a friend choosing between the two setups.

4-2 The Tutoring Service. [LO2, LO4] A student runs a tutoring service, charging $150 per session-package. Variable costs per package: materials $15, hourly tutor pay $75. There are no fixed costs - tutors work only when booked. (a) What is the contribution margin per package? (b) At 12 packages a month, what is profit? (c) If volume doubles to 24, what happens to profit? (d) Explain why the P&L lens and the cost behavior lens tell nearly the same story for this business.

4-3 The Burrito Cart. [LO2, LO4] A burrito cart sells bowls for $12. Variable cost per bowl is $5 ($3.50 ingredients, $1.50 part-time labor). The cart lease and equipment run $35,000 a year. (a) Compute the contribution margin per bowl. (b) Compute the break-even quantity. (c) Compute profit at 3,000, 5,000, and 8,000 bowls. (d) Compare: why do the two lenses diverge so much more here than in the tutoring service problem?

4-4 Reclassify the P&L. [LO3] Take the income statement from “The income statement: where did the money go?” earlier in this chapter. For each line - COGS, Selling, G&A, D&A, Interest - identify at least one component that is likely variable and one that is likely fixed (where both exist), and state what you would need to know about the business to classify it properly.

4-5 The Theater’s Leverage. [LO4] The 100-seat theater from the operating leverage section charges $10 a ticket, pays $1 of variable cost per ticket, and carries $540 of fixed cost per night. (a) Verify that break-even is 60 seats. (b) Compute DOL at 61, 70, and 95 seats sold, using either formula. (c) Explain, without recomputing, why DOL at 61 seats is enormous while DOL at 95 seats is modest - what changed, given that every ticket contributes the same $9? (d) The manager is choosing between two Friday films with the same expected attendance of 65: a steady draw of 63-67, or a risky one that could land anywhere from 50 to 80. Use the margin of safety idea to explain what the leverage math says about the risky choice.