In this lecture on cost analysis, we begin by exploring cost-volume-profit analysis and its role in strategic decision-making. We introduce the concept of total cost, including fixed and unit costs, alongside revenue to understand profit. We caution against indefinitely applying linear relationships, though they simplify economic realities. We clarify definitions of fixed, marginal, and incremental costs, highlighting potential inconsistencies. We emphasize the alignment of tangent lines with functions indicating similar slopes, reinforcing the relevance of calculus.
The lecture delves into estimated fixed and variable costs within a relivant range, step costs, and semi-variable costs. We focus on the impact of synergy on average cost in multi-product firms, challenging traditional economic teachings. The session concludes with a demonstration of incremental cost using Excel and Python, emphasizing the practical application of structured frameworks for uncertain situations. We encourage students to navigate uncertainty, showcasing the value of hands-on tasks in decision-making. The lecture underscores the importance of understanding and applying calculus and mathematical tools in managerial accounting.
In this lecture on cost analysis, we began by exploring cost-volume-profit analysis and its role in strategic decision-making. The concept of total cost, including fixed and unit costs, are introduced alongside revenue to understand profit. Linear relationships, though simplifying economic realities, were cautioned against for indefinite application. Definitions of fixed, marginal, and incremental costs were clarified, highlighting potential inconsistencies. The alignment of tangent lines with functions indicated similar slopes, reinforcing the relevance of calculus. The lecture delved into estimated fixed costs within a linear range, step costs, and semi-variable costs. A key focus was on synergy’s impact on average cost in multi-product firms, challenging traditional economic teachings. The session concluded with a demonstration of incremental cost using Excel and Python, emphasizing the practical application of structured frameworks for uncertain situations. Students were encouraged to navigate uncertainty, showcasing the value of hands-on tasks in decision-making. The lecture underscored the importance of understanding and applying calculus and mathematical tools in managerial accounting.
In the labyrinthine exploration of cost analysis, the lecture meandered through the intricate dance of total costs, fixed expenses, and revenue, unveiling the nuanced relationships that underpin strategic decision-making. Linear models, while momentarily alluring, were cast aside as insufficient vessels for the complex seas of economic realities. With a nod to calculus, the lecture unraveled the enigmatic alignment of tangent lines and functions, revealing a symbiotic dance that transcends the ordinary. In the realm of estimated fixed costs and the whimsicality of step and semi-variable costs, the narrative wove a tapestry of financial intricacies, beckoning the audience to navigate the uncertain waters with the pragmatic tools of Excel and Python. In the finale, the lecture echoed the sentiment of David Foster Wallace, urging students to grapple with uncertainty, a symphony of chaos awaiting the analytical mind.
In the cost lecture, we explored profit and loss, dissecting the simplicity of fixed and unit costs. Linear models were dismissed, deemed inadequate for the complex dance of real-world economics. The alignment of tangent lines and functions, a subtle calculus waltz, revealed the intimate connection between slopes and financial truths. Amidst the linear range, fixed costs, both estimated and true, emerged as silent protagonists. Step costs and semi-variable costs were introduced, offering a glimpse into the rhythm of financial intricacies. The narrative concluded with a practical demonstration, inviting students to confront uncertainty with the stoic tools of Excel and Python. In the spirit of Hemingway, the lecture echoed the call to navigate the uncertain seas, a voyage embraced with analytical fortitude.
Today, we’re delving into the nature of costs in this course. Specifically, over the next few lectures, we’ll explore how costs respond to business decisions. Why is this important? Well, as mentioned last time, costs involve sacrificing resources. When making strategic decisions, we need to understand what we’re giving up to pursue our goals.
This brings us to cost-volume-profit analysis, a term accountants use for the interaction between output decisions, costs, and profit. We determine how much to produce, which ties into the fundamental idea that profit equals total cost minus revenue.
Total cost includes fixed costs and the sum of unit costs for each item produced, multiplied by the number of units. Revenue, on the other hand, is the price per unit multiplied by the total units for all items sold.
Now, let’s simplify this relationship. Profit equals revenue minus total cost. When we talk about volume, we’re referring to the number of units. Companies like Apple base production decisions on projections of how costs and revenues will respond.
In essence, managerial accounting revolves around connecting business decisions to expected outcomes through various relationships. I’ll share these slides on my website, so no need to take pictures. Today, we’ll focus on modeling the cost aspect of the equation.
Now, onto an example with Apple because, well, why not? The graph depicts a linear relationship between units produced and dollars, representing both cost and revenue. Below total costs, you incur a loss; above, you make a profit—a basic cost-volume-profit relationship.
Linear relationships, however, are rare in real life. They simplify complex economic realities, and assuming linearity indefinitely leads to unrealistic outcomes, like infinite profits, which is impractical.
So, while linear relationships may hold in some instances, they can’t universally represent all aspects of reality. This understanding is crucial to avoid the pitfalls of applying linear models recklessly.
For your reference, let’s discuss fixed costs. These are costs when the output is zero. It’s essential to note that the cost at zero output is distinct from costs that don’t vary with output, although both terms are sometimes used interchangeably.
The textbook itself can be inconsistent in its definitions. Fixed cost is initially defined as the cost at zero output, but shortly after, there’s a mention that fixed cost is not necessarily equal to the output at zero. Conversations about fixed costs often involve different interpretations, so it’s crucial to clarify the context.
Similarly, terms like marginal and incremental costs can be defined precisely but are frequently misused. Marginal cost refers to the cost of producing the marginal unit, while incremental cost is the cost of producing the next unit. Although these are often the same, there are instances where they differ. The textbook might define them in a certain way but later contradict itself.
If colleagues or employees use these terms inconsistently, it’s advisable to understand their intended meanings and provide clarification. Moving on, marginal cost is defined as an instantaneous rate of change, while incremental cost is the changing cost over a specific increment (one unit increment).
Variable cost encompasses all costs that vary. Marginal and incremental costs are fixed when they are zero, but if not, they vary with output. Average cost is the total cost of producing the output divided by the number of units produced.
In the context of a single-product firm, average cost is straightforward, but it becomes complex in multi-product firms. This complexity is often overlooked in introductory managerial accounting.
Entering a market is determined by whether the price is greater than the expected average total cost—a critical factor in introductory economics. However, in managerial accounting, real-world scenarios introduce complexities that challenge these simplified models.
Cost objects are entities for which costs are measured, and cost drivers are variables associated with these costs. Both can be outputs or other factors influencing costs.
Linear cost relationships are often assumed but can be unrealistic. The graph presented illustrates a nonlinear cost relationship. The area to the left of the line (X to A) represents setup costs when initiating production. The area between X to A and Y to B reflects the business-as-usual range, where marginal costs stabilize. The area beyond Y to B signifies additional setup costs when venturing outside the designed range.
Understanding these economic significances is crucial for making informed decisions and avoiding pitfalls. Linear models are useful within a certain range but may not apply beyond, requiring careful consideration when planning for growth or changes in production.
Let’s revisit these slides. This new slide appears quite similar to the previous one. Now, on this slide, we’re dealing with the same function, but we’ve introduced some of the cost terms we discussed earlier.
We have fixed costs, which occur at zero, and then there’s point C, chosen somewhat arbitrarily for interest. Note this: the marginal cost at C is the slope of the tangent line at C.
If you recall calculus – and I hope you do – the marginal cost at a point is the slope of the tangent line to the function, which is essentially the derivative. I’m not asking this to make calculus sound fun; I get it, calculus can be challenging. However, understanding derivatives is crucial. It’s not about inventing calculus; it’s about using it, and it serves us well here.
So, despite the potential unpleasantness of calculus, let’s acknowledge its existence and its usefulness in this context. Marginal cost represents the slope of the tangent line, while average cost can be seen as the slope of the line from O to C – essentially, it’s C divided by Z.
In simpler terms, average cost is the total divided by the number of units, a fundamental concept in averages. Now, let’s delve into a question. If marginal cost is the slope of the tangent line and incremental cost is the cost of the next full unit, when do they coincide on this graph?
Feel free to answer; if you don’t know, that’s fine too. It turns out that the slope of the tangent line matches the incremental cost when the tangent line aligns closely with the function. This happens when the tangent line overlays the function, indicating that the slopes are the same within a relevant range.
I hope this clarifies it a bit. Let’s move on to the next point. This straight line approximation further emphasizes the idea. The equation for this straight line is, in fact, the tangent line for most of the relevant range. Calculating them would likely yield very close values in the business-as-usual scenario.
Now, regarding fixed costs, what we have here is an estimated fixed cost. The true fixed cost is at the bottom when the factory is closed, and we’re incurring expenses. The estimated fixed cost, in this linear range, represents the startup costs.
Lastly, I’d like to introduce step costs and semi-variable costs. Step costs imply that many expenses are incurred in finite chunks, not continuously. On the other hand, semi-variable costs involve fees or payments to access or acquire something.
I hope this breakdown helps you understand these concepts better. Let me know if you have any questions or if there’s anything specific you’d like to explore further.
Let’s revisit these slides. Oh, this is the new slide, and it looks quite similar to the previous one. On this slide, we have the same function, but now we’ve incorporated some of the cost terms discussed earlier.
We start with fixed costs, which occur at zero. Then, we identify point C, an arbitrary but intriguing point. Note this: the marginal cost at C is the slope of the tangent line at C.
If you recall calculus, do you remember how the marginal cost at a point is the slope of the tangent line to the function? It’s essentially the derivative. I’m not asking this to test you on calculus; derivatives are just incredibly useful. Calculus may seem challenging, especially with integration, but a simple derivative, like the one here, is intuitive and straightforward. Most textbooks usually begin with integral calculus, but our goal is to apply calculus, not invent it. Despite the challenges, calculus is a valuable tool for us.
So, marginal cost is the slope of the tangent line. Additionally, think of average cost as the slope of the line from O to C, which is essentially C divided by Z. In simpler terms, it’s the rise (cost) divided by the run (number of units), which aligns with the concept of an average.
Now, let’s address a question. If marginal cost is the slope of the tangent line and incremental cost is the cost of the next full unit, when are they the same on this graph? Feel free to say you don’t know; that’s perfectly fine. The answer lies in the tangent line aligning with the function. When the tangent line overlaps or is very close to the function, the slope of both becomes the same. This happens in the highlighted area.
To illustrate this further, the straight line approximation serves as the tangent line for much of the relevant range. Even though fixed costs are estimated and not the true fixed costs, this estimation is acceptable within the linear range. It’s essential to recognize terms like step costs, which involve payments in finite chunks, and semi-variable costs, where there’s a fixed fee and additional costs based on usage. Although reality might involve discontinuities and non-smooth lines, for simplicity, we often assume continuity.
Moving on to the homework assignment on cost in a multi-product firm, we have three firms with two products each. The functions involve linear and squared terms, and one with an interaction. The first step is to fill in the table for total, average, marginal, and incremental costs by plugging the numbers into the cost functions.
This spreadsheet simplifies the process, and locking specific cells allows for easy modification. This skill of quickly setting up and manipulating formulas is valuable when dealing with numerical questions.
In this course, one key takeaway is the ability to navigate situations when you’re uncertain about what to do. When faced with pressure and uncertainty, a helpful approach is to engage your hands in a task while your brain processes the situation. By creating a structure, such as a notebook with data loaded in pandas or Excel, you can seamlessly transition to analyzing the data once you’ve calmed down.
Setting up a structured framework allows you to take a breath and invites you to take the next analytical step. While Python is introduced for its ease and efficiency, using Excel is completely acceptable. Python can be advantageous for certain tasks, but it’s essential to understand that the choice between Python and Excel depends on the complexity and maintainability of the task. For example, Google Colab can be used for Python, where pandas and numpy facilitate efficient data operations. The process involves creating a dictionary, converting it into a table, and Python’s concise syntax makes these steps straightforward. Moving on to cost functions, defining them as functions in Python involves using “def” and specifying the variables, making it a practical and fast process. The ability to pass quantities to a function and receive results enhances the analytical workflow. Average cost is discussed with a reminder that synergy between products affects how costs interact. Average cost becomes misleading when synergy is present, emphasizing the importance of understanding how products influence each other in cost systems. Marginal cost is introduced as the derivative of the cost function, emphasizing the relevance of calculus. Python’s capability to handle such calculations efficiently is briefly mentioned. The concept of synergy is further explored, highlighting its significance in real-world scenarios and decision-making for multi-product firms. This challenges traditional economic teachings about average total cost. The session concludes with a demonstration of incremental cost using Excel. The incremental cost is explained as the cost of one more unit of a product minus the original cost. In Python, this process is simplified, reflecting the language’s power in handling mathematical operations. Unfortunately, due to technical constraints, the promised demonstration of 3D scatter plots is deferred, but a walkthrough of the process in Excel is provided. The session wraps up with a reminder of the upcoming assignment and a motivational note about the value of navigating uncertainty.
Okay, nature of costs.
So we are interested in this course in general, but in particular today and over the next couple lectures, couple, three, four lectures, in how costs respond to business decisions. Why are we interested in this? Because, as we mentioned last time, costs are sacrifices of resources. So when we make a choice about something we want want to do, some strategic decision, we need to know what we have to sacrifice in order to do it.
This leads into – oh, we need to know what we have to sacrifice. We also need to make plans to make sure we have the appropriate resources at the time we need it. This leads into something called cost volume profit analysis.
analysis, which accountants love to give fancy names for very simple things that didn’t need a fancy name. But they call cost -volume profit analysis is just the interaction between output decisions, how much are we going to make, how much are we going to sell, costs, and profit. So we’ll decide how much to produce.
I think I should just cut to the end. the next slide and then that will, yes, okay. So, the fundamental idea here is, I should say this is going to be a minor theme today and something we’ll talk about more in the next couple of lectures.
But, so profit is just simply equal to total cost minus revenue. Total cost is fixed cost plus the sum of all of the cost per unit for each of the things that we make times the number of units. And then revenue is just price per unit times units summed across all the different things we make.
So this is just to just kind of refreshing your minds that there should be a plus sign here. here, total cost plus some. There we go.
So this is just to kind of refresh in your minds what the cost volume profit relationship is. Cost here is easy because cost is in the words, right? This is what the cost is. Profit is just the difference between cost and revenue.
That’s also backwards. Not quite as embarrassing as my laptop, but nonetheless embarrassing. Okay.
Plus, no. I’m my own worst enemy today. Okay.
Revenue minus total. cost, total cost is equal to fixed cost. Okay, so you can see from this profit and cost in the cost -volume profit relationship, the volume, when we talk about volume, we’re talking about the number of units.
So when I say that we’re interested in how costs respond to business decisions, for example, Apple chooses how many units. devices to produce based on projections of how costs and revenues will respond. This kind of simple sentence about a decision that a company will make is the basic description of most of what managerial accounting is.
We’re trying to connect a business decision through some set of relationships. relationships to some expected outcome. Again, I’ll post these slides to be posted.
Don’t worry about taking pictures. You can take pictures, but I’ll put them up on my website. Let me just show you where.
website under courses and this is if you follow the link to my course page you’ll come follow the link in canvas to the course page you’ll land here and then down here this is the slides from Tuesday and then this is the Excel sheet Okay, so today we’re going to focus on the modeling cost portion of the equation. So, of the equation from the last slide. And all my examples are about Apple because, sorry, I just did that.
Okay. I have this, this slide. of weird situation where the classrooms I teach it all have different projectors with different DPI’s so I have to adjust some of the images so that they are readable in different places and I’m trying to get it right so let’s try 1 ,500 here, okay Is this somewhere? readable? A little bit, a little bit.
Okay. So this is a simple example of the equation we were just talking about where everything is linear. We have this unit produced on the horizontal axis, the x -axis, and the dollars on the y -axis.
We call it dollars here because it’s tracking both cost and revenue. revenue. This line, the blackish line, I’m somewhat colorblind so I’m going to say color sometimes and you might be like, “That’s not the color that’s there.
Sorry. That looks black to me. This one looks a little bit blue to me.” So total revenue is this line.
Total cost is this line. Unsurprisingly, I’m sure by this time in your career as students. students you know that when total revenues are below, total costs, you’re making a loss, and when you’re above, you are getting profit.
So again, the simple bottom of this image is just the simple basic cost -quality profit relationship from before. Okay, can we see anything? – sorry, they keep fiddling with the size of these. Okay, so is there anything about this graph, this linear graph that is unrealistic? I’m having a hard time getting the – this portion can be – everything can be legible at the same time using this.
Okay, so what is unrealistic about this? and that is, I can really read that. Gee whiz. Wong Ka Wing.
Is anything unrealistic about this graph? Yes. Yeah, yeah, yeah, real life is a linear. Yeah, you’re totally right.
You’re totally right. Yeah, totally right. So these are linear, linear, linear relationships are rare.
100%. Yes. Something else.
There’s more. There’s more. There’s actually much more.
Sorry to shrink this again. again Chelsea chew yes anything else unrealistic about this graph that’s okay that’s okay that is just as good an answer as any other answer because I think So one of the answers is Everything’s linear. The other answer is I don’t know that’s actually really correct as well because This is just a picture.
We don’t know what the underlying reality is. Maybe linear is true Maybe linear is false. We don’t know so I I actually that’s like actually a really good answer.
We don’t know what’s unrealistic because we don’t know the underlying economics of the firm. Now, the last thing that I want to point out is it’s unrealistic about this. is that what happens if we keep increasing units produced forever? What happens to profit? If units produced goes to infinity, what happens to profit? It goes to infinity! That’s impossible! That can’t happen! right? You can’t have infinity of anything.
You can’t have infinity of money. Right? There’s a money supply, right? Money’s a real thing. Even if it just is in like somebody’s bank account somewhere, because at some point the bank account gets big enough that to hold the digits you would need bigger computers.
So how much of computers would you need to hold an infinity bank account? Infinity of computers! This says that someone needs infinity computers to keep track of how much money they have. That’s crazy! Sorry for yelling, but one of the things that’s crazy about linear relationships is that if you don’t stop applying them, that, insanity occurs. That’s the big takeaway here.
At some point, no matter how true linear relationships are in some little portion of reality, there are portions of reality that cannot possibly be represented by a linear relationship. relationship. OK, so let’s look at some cost terms in the context of a single unit firm.
Again, I apologize that these don’t quite fit the screen. I’ve been trying to get the right amount of tax money. each one.
But this will be nice for your reference. So we have fixed cost, this is the cost when output is equal to zero. Something to keep in mind is that cost at, at zero output is a very specific thing.
It’s slightly different from costs that do not vary with output. These are two slightly different things. They have important differences that we’ll point out in a second, but fixed cost gets used to mean both things.
In fact, if you pay close attention to the textbook, you’ll notice that he says fixed cost, he’s cost at zero output, and then really, really quickly thereafter, he actually draws a line where fixed cost is not equal to the output. at zero. So notice– I just want to emphasize the fact that these things are inconsistently written in a textbook, but everyone makes these– every conversation you have about fixed cost, someone is going to just make a decision about which way they mean it.
And they may not be explicit about what they mean. So you may have to listen and figure out. or ask the follower questions and say, “Hey, do you mean fixed costs, like, actually at zero, or do you just mean within the range of out what we are considering, this cost does not change?” Okay? Um, marginal and incremental costs are another great example of a term that people will define precisely, and then immediately misuse.
misuse. So the marginal cost is the cost to produce the marginal unit. Incremental cost is the cost of producing the next unit.
These are often the same thing, but they’re not always the same. And I’ll show you an example where it’s not true. And again, this is one that I believe the textbook defines one way and then immediately contradicts itself.
Now, if your boss starts using these inconsistently, probably just try to figure out what your boss means and just go with it. But like if somebody that works for you starts using these inconsistently, you should definitely help them out. Ok, so, the marginal cost, like the most pure cost, definition of marginal cost is something that Will be a little bit familiar to you from calculus.
Don’t worry. We’re not gonna do much like calculus We’re gonna acknowledge the existence of calculus in this class, but we’re not gonna like Do the painful things about me? okay, I’m like not super good at the mathematics that you have to do in upper division mathematics, but like we’re just gonna know the calculus exists but don’t worry, don’t worry. Anyway, the marginal cost is an instantaneous rate of change whereas incremental cost is the changing cost over a specific increment, one unit increment.
In fact, lots of people misuse the word increment, they use it to mean all sorts of of things. “Increment” means a defined distance. So, like, one unit is a one -unit increment.
A lot of people use it to mean kind of just more. In fact, many, many kind of math -related terms are used by the general public to just mean more, like an exponential increase. Most people just mean big changes.
increase or increase I don’t understand or I’m scared and want everyone in the room to think I said that’s a smart thing. So incremental cost is an increment, marginal cost is an instantaneous rate of change. Variable cost is just, all costs vary without them.
So marginal and fixed costs when they’re zero, sorry, marginal and incremental costs, if they’re zero, are fixed, but if they’re not zero, then they, by definition, vary with output, so their very average cost is the total cost of producing the output over the number – sorry, if I read that, I’m just going to keep saying it – oh, also, don’t – it’s okay if your phone goes off in class. Totally okay. because the first time I taught during the PhD program this is like in 2013 I said I’d never taught the person I did teach in person once in 2013 so in 2013 I was like turn off your phones hide your phones I was really really strict about it I didn’t let anyone have their electronics out and then because I was using my laptop attached attached, my wife called me, who’s the only person who can actually just call me.
And I silenced, it was my phone, I silenced it, and I sent the quick reply, like I’m in class, just by tapping the screen. And I was like, whoo, nobody noticed it. But I had my laptop attached, and it broke throughout my laptop, and that was hooked up to the speakers for the classroom.
And it was like going off really loud, and everyone was like diving in their bags and stuff, and finally had to be like, “Sorry, that was me.” The worst thing about this is I was not presenting. I had invited an accountant to present from one of the big auditing companies in Salt Lake City. So I was the most embarrassing person in that room.
And so for that reason, if your phone goes off, that’s it. just fine if you want to like actually talk on the phone but otherwise just don’t feel bad if your phone goes off because I have done worse okay average cost it’s just total total cost of producing the units you produced divided by the number of units you produce now notice we’re talking about a single product firm and here, and if you remember from the homework, which I did two last night, is average cost gets complicated in multi -product firms, and that’s kind of the point of why I wanted you to see that, because so often in introductory managerial accounting, we just say average cost is total cost over number of units produced, dA, and then we’ll write down, and then we’ll say, oh yeah, but real firms have multiple products. And we just move on.
So that’s why I wanted to struggle with that a little bit. I wanted you to think about it. And we’re going to talk about it more later.
Also, if you remember from your introductory economics course, what is the piece of information you need to decide whether or not to enter a market? It is whether or not to enter a market. the price is greater than your expected average total cost. Alright so this is a critical in the world of economics, the introductory economics, this is one of the most important pieces of information and in the world of introductory managerial accounting it is a very simple piece of information to produce but those worlds are so simplified that I want to show you that they just blow up and become completely unused for our context.
So, that’s a little cliffhanger. We’re going to come back to it. Cost objects, I always get confused by these terms.
Cost objects are just the thing we’re trying to put associated costs with. We’re trying to measure costs for a cost object. A cost driver is, it’s really the X variable in most of our plots.
A cost driver is just the thing we’re trying to put associated costs with. going to change and try to figure out what costs What? So it’s the thing that we think is driving the cost All of these things can be output, but they can also be other things that output that’s why I wrote an activity or item and a factor or activity in Managerial accounting we often talk about things as activities that are like outputs or inputs that’s okay that’s the the term list again I’m gonna post this I’ll post it tomorrow on my website maybe today if I have time but Okay. Now, we talked about why a linear cost relationship is unrealistic.
What I have here is a nonlinear cost relationship. And I want to point out that this, if you just, like, we always, we always assume in the way we talk about costs that the relationships between units produced and cost are linear, but our mental model should be something like this. And so now I want to talk about one.
Oh, and I want to use the term economic significance because that confuses some people and so I want to find it out, so let’s see if we can use this. Okay. Okay.
So. the first question is, what is the economic significance of the area to the left of the line from x to a? So that’s this little challenge. What’s the economic significance of this? And I need to– oh, no.
Sorry about this. OK, this is for– for Lanza manual sun. Okay.
So, the question is, what is the economic significance of the area to the left of the line from next day to this? » Could it be that when you’re setting something up, when you’re starting a production, you have also set up costs of – » 100 percent. » Yes. that’s exactly there’s a ton of like we could we could brainstorm a ton of reasons why this is true but you’ve got it this is the overarching idea that the you know the fixed cost this is like the rent you have to pay when nothing’s happening in your factory and then just like you said that so you have to get the first person to come in that first person comes in and there’s like one unit of production but you have to pay for the whole person And then that person makes two units and that’s a little less expensive It’s so pretty expensive and then it’s like it starts to wind up Yeah, and you’re right if you think of this as like Starting a factory from day one is zero output Then you have to do a bunch of stuff hire the first worker rent the first machine Or if you’re a company that like does seasonal work work, this is just when everything’s idle and then when you start up production.
Exactly. All right, let’s ask the second question. Okay, Chen, Chen.
Yin -Kuan, what is the economic significance of the area between X to A and Y to B? So this area, what’s the economic significance of this area? I mean, this part of the line, not this area, this part of the line. line. Yes.
Yeah. Yeah. So the answer is that this is really that the slope of this section is sort of the business as usual marginal cost.
So what Lance said is that here we’re starting everything up. This is kind of the business as usual area. So over here the marginal cost is changing with every unit.
And here the marginal cost is changing with every unit. stabilizes across units, or almost stabilizes across units. So here the slope is basically the marginal cost.
It’s probably also the incremental cost as we’ll see in a minute. But yeah, this is sort of the business as usual. This is the range where you’ve sort of designed your system.
to function efficiently. Often we’ll refer to this as the relevant range, because if we make a decision to move just a little bit in this area, increase production a little bit, we can probably predict based on what we did here, what we did there. So this is a really easy area, a really predictable area, for us to offer.
Okay, then the next question is what is the economic significance of this area? And by the way, this total cost label is just referring to this blue line is total cost -adjustable. So the area over here. what is that part of the line doing? What is the fact that that line shoots up? What is that? And now let me pull up.
Yipchuk Kong. Yes. Yes Yeah, yeah, that’s perfect.
It’s a great answer great answer So it’s exactly right that basically everything that Lance said had to happen here Has to happen again when we get out of our area of of where we designed the firm carefully to function here. And that area is just, now we have to buy a second factory or raw materials become scarce, that sort of thing. So this is actually a really, really interesting question and part of the answer, part of the reason why even if revenue is linear, we don’t get profits that go to infinity.
infinity because it’s very, it’s, while it’s unrealistic for profits to go to infinity, it’s very realistic for costs to at least be asymptotically infinite. So you wouldn’t actually pay infinity for something because you don’t have infinity resources to, to sacrifice, right? But you get to a point where the costs are so high that you would have to sacrifice all your resources now remember when I said plans and goals and managerial accounting are Precise sorry goals and targets are precise plans. They are not aspirations This is part of what? If your sales team if you’re sitting here, and your sales team is like this year.
We’re gonna sell everything and then they sell infinity units your company burns to the ground. Because cost goes to infinity you have to sacrifice all your resources and you go bankrupt before you can get paid. So this is why if you’re going to sell a ton of stuff you have to design a firm that can actually sell a ton of stuff.
So if you want, it’s a pretty good idea. relevant because of the influence of venture capital. Venture capital business model is invested in a bunch of stuff and hope that one of them returns 100, 200, 1 ,000 folks.
That means that venture capital firms are aiming here two problems. One, this is the hardest part of the graph two per day. Two, that is the most expensive part of the graph.
It’s very difficult to design a firm as a proof of concept and have it function efficiently at implementation and be able to scale infinitely. This is partially why venture capitalists function so well in software because software does scale well. But think about Uber.
Uber is worth it. of the best venture capital stories. In fact, the reason I keep remembering Lance’s name is because Lance Armstrong, the famous American bicyclist, is wealthy not because of winning bike races, but because he bought Uber at the, he was like one of the venture capitalists that funded Uber.
But Uber is not profitable. Why? Because it’s really hard to scale. Their costs keep increasing, and they don’t even have any cars.
So I think that the point of me yelling about this graph and what’s been conveyed in all of your questions is one, that this is a really good approximation for how companies work. And two, if we’re talking about a linear relationship between costs and outputs, or sorry, costs and outputs, we have to make sure we’re talking about plans that fall here. As soon as we’re talking about plans that fall there we have to know that our linear models don’t apply.
Now of course we’re going to use linear models because there’s so much easier to use, so much more reliable to create, they require like really specific clear assumption, we just have to make sure that when we do linear stuff we stay in here or make plans to build a new structure. This is one of those moments when I ask you hard questions in the homework, and then I read a program that asks me really easy questions and I answer them wrong, so then you can all kind of like laugh and be like, hahahahaha, he asked himself an easy question. it failed in front of all of us.
Let’s go back to these slides. Oh, this is the new slide, it looks so similar to that. Okay, so on this slide, we have the same function, but now we have some of the cost terms we talked about before.
So we have fixed costs, which happens at zero, and then we have this point C, which is just an arbitrary point. We picked it so it would be slightly interesting, but notice this. Marginal cost at C is the slope of the line tangent at C.
If you remember calculus, do you remember calculus? What’s the marginal cost at a point of the slope of the line tangent at C? of a line tangent to the function? It’s the derivative. I didn’t ask you to answer that question because I wanted to. Derivatives are so much fun, right? Everyone is like, calculus is so hard.
And calculus is pretty hard. Integration can lead to really nasty algebra and B -variance lesson. But a simple derivative like this is just like– the intuition is really good, it’s super useful, calculating it is pretty straightforward, right, like you just take the exponent, multiply by it, reduce the exponent by one hour, like it’s very, very simple, they should kind of leave with this, like most textbooks start out with integral calculus, but I see the reasons, like that’s how we kind of came up with the intuition for calculus, but we’re not inventing calculus, we just want to use calculus, and this is great use for calculus.
So even though calculus can be unpleasant, we can just acknowledge that it exists and it helps us here. So marginal cost is the slope of the line tangent. And then you can think of average cost as the slope of the line from O to C, which is just a fancy way of saying it’s C divided by Z.
I mean that’s exactly what the slope of this line is. Rise of the run, rise of the cost, runs the number of units. And it really like, this is just what an average is, right? An average is the number of things, or sorry, the total divided by the number of things.
have to jump back into the notebook. Okay, here’s the question. If marginal cost of the slope of the tangent line and incremental cost is equal to– if marginal cost is the slope of the tangent line, incremental cost is the cost of the next full unit, then when are they the same on this graph? And the question is for Fon, the yam saying that you don’t know is just fine because I’m also here to tell you so.
But when is this this slope going to be the same as the incremental cost? Does the, does the question make sense? When is the slope of tangent line the same as the incremental cost? At zero? That depends on the function, but on this that’s not true for this this line oh Also, is it you that I asked the question to oh? So it’s actually going to be the case. Oh, can I just ask is Is this person here? You’re here Okay. Do you have an answer? It’s okay.
The answer can be. Okay. Okay.
That’s a great answer. That’s a great answer. The answer can also be stop bothering me and tell me the answer.
That’s a good answer. Yeah. So it’s going to be, see how this line is like almost lining up.
up with the line of the function? And then if we go over here, it’s not, right? It’s going to be further away. It’s going to be when the tangent line actually lines up with the slope within the relevant range. That makes sense all kind of, maybe if I…
draw it so like we have this here the tangent line is a big difference here oh I drew that poorly here the tangent line overlays the function and when the tangent line is overlays the function, then the slope of the function itself and the tangent line are the same. And the incremental cost is the slope of the function itself over a one -unit interval. And here, since the one -unit interval and the tangent line fall on the same line, or like really, really close to the same line, then we can save the diversity.
the same. And we’ll show some examples of that when we work through the problem. Okay.
Now, this straight line approximation illustrates the point that I was just making even more solid. This straight line approximation equation. is actually the tangent line for most of this kind of relevant range area, right? For the business as usual area, these are close enough to each other that if we were to calculate it, we would probably end up with the same number.
We’d end up with really close to the same thing. So the answer to my question is this area. This is where they’re the same.
They start to differ. when they take off in different directions. Second thing I want to point out here is, see, this fixed cost, this fixed cost is an estimated fixed cost.
It’s not the true fixed cost. Because the true fixed cost is down here, when the factory is closed and we’re just paying money. So this estimated fixed cost is that in this linear range– range, this level of costs don’t change without [INAUDIBLE] So it’s all the startup stuff that Lance mentioned.
And because you don’t have the same name as a bike racer, do I know? I don’t remember your name. But up here, these areas have different sets of fixed costs. Costs can be even differently there.
But here, we can estimate fixed costs. as being whatever this level is. And so variable cost is really just, in this estimation, it’s the portion in our linear approximation or straight line approximation that does respond to output.
Then the last thing I want to put up here, this is just… I want you to recognize these terms when they come up. Step costs just means that when we’re saying everything is linear, most things you have to pay for in some sort of finite chunk.
Like a few things, like maybe your power bill, you pay mostly like on a decimal basis. you have to pay exactly as you use everything. But, you know, factories, you have to increase the factory and, like, increments.
A lot of computing stuff that you’re doing, like, through Amazon Web Services would be more like semi -variable. But most costs have some aspects of steps where, you know, if you zoom in tightly enough, you’re really jumping up over increments rather than… than on a continuous basis.
And then semi -variable costs are like, most things, you have some sort of fee you pay to get in, you have to buy something and then you have to offer it, you have to rent something and then you have to offer it. So a lot of the time costs aren’t gonna go to zero at zero. These are just two things to keep in mind, but in reality, most of the time, time, lines don’t go through zero, and most of the time, they’re not actually smooth, but we’re going to go back to pretending that they’re kind of at least continuous, right? This is discontinuous.
Flat. Jump. Flat.
Jump. We’re going to draw a lot of continuous relationships. Also, estimating continuous relationships is a lot easier if you don’t know the structure of the data.
than estimating discontinuous relationships. Because you have to, like, first have some way of either exploring the data or having some algorithm that tells you, ah, this is actually discontinuous, and here’s where it might be, and it gets to be a whole thing. Okay, so now let’s talk about the homework assignment.
And… because I’m still You talked a little too much About the stuff up to this point. Okay, so cost in a multi -product firm We have three firms and they all have have two products, Q1 and Q2, that’s going to be a quantity of product one and a quantity of product two.
This one is just two linear relationships, these both have a square term. An early version of a PDF that I posted was missing this square, which I corrected like several days ago, but I think if you looked at the assignment … right away, it walked in the incorrect version.
So I’m sorry if some of you saw one without a square term. It’ll be fine. If you answered it without the square term, you’ll still get incorrect, because it will answer quite a slight question.
Okay, and then the last one has two linear terms, and then one term where it’s two things multiplied by each other. We’ll talk about that in a minute. but this is what we call an interaction.
These two things interact. Okay, so the first step of the question was to just fill in the following table on total average marginal incremental. So what do we do well the first thing is Start with our cost functions, and we just plug the numbers into the cost So let’s do that in in here, so this is just a Excel spreadsheet where I’m going to just write the functions into these cells.
Now, one, because we’re a little pressed for time and two, because typing in front of a large group of people inevitably ends up with mistakes. I have written down the things that we want to type. So we’re going to type go to the blank workbook, come to total cost, and I’ll just paste it in there.
So I’m typing in the formula, 10 times A4. And in Excel, this is kind of like a variable, right? We’ll just refer to the cell, and each cell, and then we’ve written down the equation. Now, notice that what it says right now is 10 times A4.
A4 times– or plus 5 times B4. I’m going to do something that I want you to see. I’m going to put a dollar sign in front of the A and the B.
In Excel language, we can call it a language, this means that that reference is
So this means that I can copy this over and modify it without changing the numbers, or sorry, without changing the letters. This is super useful because now I can just double click on the corner. and It’ll fill down And it just took this Like if we look there, it says a four five six seven So that’s average cost for firm one.
We can do something similar for firm two Just come grab formula, paste it in here, oh, I have to actually grab the formula, okay. so now we have the formula, we have A’s, B’s, these are all in row 14, notice here I put in a little carrot, the carrot raises us to the second power, drop it in, double click the corner, and we’ve got our total cost, same thing for firm 3. okay, so here the formula is just 7 times this cell, 9 times this cell.
cell, and then these two cells multiplied by each other. Okay? Total cost is pretty straightforward. This is a good – just being able to do this confidently and quickly, like anytime you – anytime somebody asks me a question where it’s just like some numbers, I want to throw it either into a spreadsheet or into a pandas data frame while I panic.
Thank you. okay? So one of the things that I want you to take away from this course is being good at knowing what to do when you don’t know what to do. And I think one of the keys is when you don’t know what to do and there’s a little bit of pressure, have something you do while your brain freaks out.
Like give your hands something to do while your brain freaks out. And so anytime I freak out about a question, I just let my brain freak out and let my hands load up. data into some sort of structure.
So then when I calm down I can start analyzing the data immediately. So this is sort of– and the nice thing about this is if you set up like a little bit of a structure in a notebook and then you put in the data, whether you do this in pandas or in Excel, then you can like take a breath and it’s like kind of inviting you to take the next step. So I think that’s quite nice.
Also this is super– this sort of– of thing is super fun. Okay. Yeah, so for total cost, we’ll just plug this into the function.
Don’t panic. Again, some people ask me, do you have to use Python? You do not have to use Python. Please use Excel.
I want to show you Python because I think… that it’s going to be useful for you to continue getting exposure to Python and because some things are really easy and fun to do in Python and just miserable to do in Excel. Like a lot of introductory things are easier in Excel, but at a certain point, it becomes just impossible to maintain and really easy in Python.
And so all of our examples are going to be in that. that easy in Excel land. But I want you to know that when you hit the wall later, you can think about switching.
OK, so I’m just going to really quickly show you how this would work in Python. If you go to Google Colab, then you can just type import pandas, pandas is like. that framework.
And you know how Excel knows how to do math across cells? That’s what pandas is. It’s like the cells, and they know how to do math with each other. And then numpy, numpy is numeric Python.
And this is just fast ways to do operations on vectors. Because one of the funny things about Python, one of the cool things about Python is you can just– be like, hey Python, this thing, give it that name, and Python will be like great. Whereas if you did this in like C++ or Rust or Go, you’d have to be like Python, or I’m sorry, you’d have to be like C++, Rust or Go, here’s a place in the computer’s memory and here’s a thing, would you please be a good chap and take this thing, put it there, and that’s a huge, and then you can make mistakes and you would lead to like, stack overflows and all sorts of madness that if we’re just trying to be accountants, we don’t need to like do that low level stuff.
You just want to be like, hey, take this number. I don’t want it, hold it, take it. So that’s what these two things do, they make it fast.
OK, and then we can just drop these things into a dictionary. I’ll post this also that you can look at it on your own. And then, yeah, so we just put it into a dictionary.
And then we can make that into a little table of data. So the main thing I wanted to show you is that it’s just a couple steps in Python to get a similar little tabular way out. Okay? Don’t worry too much about the steps.
I put them in the– I posted them. For those of you that took the intro to Python class and got like obsessed with Python or introduced, I forget what the name of it is. But for those of you that took it, had a lot of fun and like it and want to do more, I’m putting all of the steps to do what we did in Excel in here.
But for those of you that are like, this is making me nauseous, that’s fine. You don’t have to do this. okay? So we did this from 1, 2, and 3, and if we ask it to show us from 2, it will show us from 2, it’s all the same at this point.
And then we have to write down our cost functions and we’ll write them down as functions. In Python, we define a function by just saying def, define, name the function, put q1 and put q2. because that’s what we’re going to put into the function And then we tell the function what to return and we return just ten times q1 five times q2 The only thing that’s a little bit of a curveball here is that when you’re talking directly to Python like this Instead of using a carrot for squaring or for exponents you do times times Not the carrot because the carrot does something else.
The caret was kind of already assigned when this part of the Python group. So same information and then we can we can see that if we pass that information to Python it gives us back the same numbers. And this is how you call a function.
You will notice it. this cost q1, q2. Now we just wrote cost q1, q2.
And it does the instructions with what we gave. This is called passing to a function. So we pass the quantities to the function.
And then there’s a kind of a slow sign. of simple way, but based on kind of the amount of chatting and glazed looks in your eyes, I think that we probably should focus on getting through everything in Excel. And then, yeah, I see some emphatic nods.
Okay, so we’ll focus on getting through everything in Excel. Okay, so total cost. We finished.
Okay. Okay, now I just need to… to – all right, so how do we do average cost? Okay, average cost.
Average cost is just going to be – we’re going to pretend that the multi -product firm is a single -product firm. So we’re going to take that cost function we had before and we’re going to take that cost function we had before. for the average cost of Q1 we’re just going to plug in 0 for Q2 and calculate the function and then divide by Q1, okay? So we just pretend we don’t produce any Q2 at this firm, what’s the average cost? Then we do the same thing for Q2, plug in 0 for Q1, calculate it.
Thank you. same thing in the next one. Let’s do these and we’ll come back to firm 3.
Okay? So what we’ll do here is we’ll just copy, oh, let’s go from 1, command C, command V. Oh, and because – oh, I’m in the wrong pitch. Okay, so let’s come here.
Madness isn’t suing. Okay, so I just copied and pasted. and noticed that because I hadn’t locked This is Okay Here we are so I’m going to go to the cell for total cost.
I’m going to copy total cost I’m going to paste total cost Into the next call because I locked the references. It’s going to be the same number Now I’m going to do what I just showed you on the previous slide, which is plug -in zero and then divide by the other quantity. So I’m deleting plug -in and zero and removing from the function the same thing.
I’m going to put parentheses around it just so that Excel doesn’t get confused, and then I’m going to click over here. here, so that I can pass that variable in. Boom, gives me back 10.
That’s good. That’s what I think it should be, and I’ll fill it down. I see some hands kind of moving up, right, but it looks like everyone’s just touching their faces, not raising their hands, so it’s okay.
Okay, and then we do it again, but here we’re just going to… delete the first term Because again, we’re pretending the firm when we calculate average cost we’re pretending the firm doesn’t produce the other product Yeah, and we filled in So here’s the average cost for firm one and I’m just going to skip to the slide where I’ve already inputted it for firm two so that we can keep moving along. But you see up here I just took the total cost formula, deleted all the terms with product two in it and then divided by product one.
And then here I took the the total cost formula, deleted all the terms that include product 1, and divided by product 2. So again, I’ll say this like three or four times, four, and we do average cost in a multi -product firm. We’re making up a pretend firm, and pretending that pretend firm only produces one product, and then we can calculate average cost.
Now, if that sounds like a kind of– of crazy thing to do, we’re depending on the fact that just deleting all of the terms that include the other products will actually give us a realistic representation of the one of the actual firms. Okay? Keep that idea in mind. Now, firm three, what happens if I come through here and delete everything – so if I want to fill an average cost for Q1, that means I have to delete the B term and this term, right? Because I’m just plugging in 0, right? What happens to something that you multiply by 0? Poof! It disappears.
Right? So, to calculate average cost, we need to plug it in zero for all the products that we aren’t trying to average. But if we do that, part of the cost, part of the, one of the terms that includes our product that we’re interested in disappears. So, mathematically, we can do this.
But we have just created a pretend firm. It does not work the same way. way that the real firm does right the pretend firm there’s no interaction between the two products but in the real firm there is an interaction between the two products this is this is a critical point and I have a slide about it so I’m going to pause for a moment to go back to the slides so what about from three? What does Q1 times G2 mean? I mentioned this before.
You call this an interaction in and so if you hear like in an economist or a mathematician talking about the interaction between two variables they’re most often referring to them being multiplied by each other in some form. Now that’s a mathy thing. That’s like an econ thing.
We’re real people. We’re accountants. So what does it mean for us? Well, this means that when there’s some synergy, they have influence on each other and now why do firms merge with other firms? Synergies, so let’s imagine there are two one -product firms and they decide to merge with each other because maybe they’re marketing there’s something good about their marketing or the firm together will be able to do something more efficiently, sell more, something like that.
They did that because of the synergy. In fact, there’s no reason to produce multiple products in the same firm unless there’s some synergy from doing them together. So this weird situation where we can’t use average cost is the only situation in which firms should have more than one product.
So you guys should be following our future. In econ 1 -0, whatever they call it, in introductory econ, they said you make your long -term entry and exit decisions based on… on average total cost and what did I just show you? Average total cost is meaningless.
If average total cost is meaningful, you made a mistake. So the only way that the entry and exit recommendations from introductory econ actually work in the real world is when you shouldn’t do it because there’s no synergy. Isn’t that crazy? You’ve been lied to.
Or maybe just clearly I’m the only one that is this excited about this. But this is a big hugen you. And so, I apologize that this might have been frustrating, but I wanted you to show up to class today and be a little frustrated with average total cost, or when there’s no synergy.
cost so that when I showed you that the answer is question mark you would either want to punch me in the face which is fine or you would be like wow average total cost is actually somewhat misleading we need to make sure that we actually understand the way products interact with each other in our cost system because in this cost system it actually means producing the other product pushes cost up so being a two -product firm here is more expensive than being a single two -product firm. From a cost perspective, splitting this firm is actually the right thing to do. Very interesting.
At least they did better because this is my job for like the rest of my life. Okay, so average cost. cost.
Questions about this? If you have– think about it. Email me if there’s more questions. This is what we mean by an interaction.
You can do this by writing a function in Python, but I’m going to skip that because we have 10 minutes left. left. Now let’s talk about marginal cost.
At a half hand raise, okay, so marginal cost. Marginal cost is just the derivative with respect to the product. The derivative of the cost function with respect to the product if we are using it.
So, don’t… worry if you’re like, I don’t know what I said this a couple of times, but don’t worry if you have bad memories of calculus. Be joyful if you have joyful memories of calculus, but all we’re going to do here is take just a simple, simple derivative, okay? So, marginal cost for firm 1, with respect to product 1, is the derivative of…
the cos function with respect to product 1. So when we do this, we lose the term that has nothing, that has no product 1 in it, and we reduce the degree, right, the degree, we reduce the exponent by 1, so it goes in. Okay? If, if you want, like, a, I’m assuming that this is an okay amount of calculus.
If I were teaching in the US, there would be, like, a student protest, and I’d be probably hung for mentioning calculus to an accountant. But you were much more intelligent than most students. And that was a very politically incorrect thing to say.
I apologize. apologize to everyone except for the Americans which is me so it’s a little more complicated for C2 because when we take the first derivative we end up with one with still a variable remainder but again it’s just the first derivative we go from a square to a square and a six times Q1 to a 6 and 2Q. Okay? That’s the extent of the calculus.
And then the interesting thing here is we end up with, in the marginal cost of Q1, we end up with Q2. And this is another reason why you can’t use average cost. Average cost is not informative when there’s synergy.
synergy, because if there’s synergy, then when we take the first derivative, we still have the other quantity, which means that the cost of producing product one is a function of the amount of product two, right? And this totally makes sense. If you have one machine and two products, they compete for the machine. Okay and so to do this in the marginal cost equation we simply write down the same formula from before so the formula we just calculated is 10, so we write the formula 10, copy it down, write formula 5, copy it down.
And we do that here, but instead of it being 10, it’s just 6 plus 2 times q1 and 8 times plus 2 times q2. And then for the fun one with the interaction, now the interaction one is pretty simple. The only thing crazy about the interaction term is that we just can’t calculate the average possible.
Marginal possible. flows right through it. So it’s just 57, 57, 17, 57, 17, 17, 19.
And you can see here that this is moving in a more interesting way. It’s not monotonic. It’s not just going down as the numbers go down.
It’s actually moving around depending on what the other one is. OK, so that’s– marginal cost. Marginal cost is relatively– like the big addition is just that we have to remember the calculus exists in order to come up with what the marginal cost is.
Now, I made fun of Americans for not liking calculus. The textbook we’re using is an American textbook. And so it doesn’t mention it like kind of uses mushy language around calculus.
And a lot of the time, when you’re trying to talk to someone who doesn’t know calculus about calculus, you say things like a one -unit change rather than an instantaneous change. So the difference between incremental cost and marginal cost becomes obscure when you don’t know calculus or when you don’t understand. that there are, you can measure change within the unit, or like change can be smaller than that.
Okay, having said that about incremental cost, let’s move to, oh, for those of you that have a little desire to see something cool and calculus, and, oh, I’m sorry, if you hate calculus. calculus and are curious about Python, I’m going to show you how to let Python do the calculus here, but because we’re out of time and many of you looked a little bit like you didn’t want to see much more about Python, I’m going to just leave it in the notes for those of you that think it’ll be cool. Okay.
Incremental cost. Incremental cost. We can make this a little bigger so you can see it.
Incremental cost is, I find it really hard to save this as a sentence in a way that’s really straightforward. But, so what this says is, the incremental cost of product 1 is the cost of product, of one more unit of product 1, which is the cost of product 1. 2, and product 2, and it’s original, minus the original level of both.
So the total cost we calculated at the beginning of the exercise, we subtract that total cost from the cost of producing one more unit, and this is useful because we’re actually plugging in units, and so we’re producing have the whole function in there and so we don’t lose the information that we were losing when we did average cost. So to do this in Excel, to do it in Python, I just write a function that is that sentence I said. This is why I love Python.
If you want the computer to do something, you just like write down the code the sentence, like write down the instructions, and then, to be honest, asking Python to do something is the same as asking like a chatbot to tell you how to do something. You’re just like, “Here’s very specifically what I’d like you to do.” And then it’s like, “Okay, here we go.” So that’s, I love Python for this step because it makes it super easy. But all going to insert two columns and I’m going to call these columns the terms that we are going to subtract from.
So this column is going to be the cost of Q1 plus 1 and Q2, and this is the cost of Q1 and Q2 plus 1. 1. And you can see that the only change I made here was I just put parentheses around the Q1 term and added plus 1.
In Python, this is super easy. Here, it’s fine. But in Python, this is just a joy to do.
OK. And then– yeah. So that’s– and I just filled that in.
And then the– actual incremental cost calculation is just this column minus this column. When I do this, am I totally blocking all of them? When I, like, stand in front of the point, I think I, with that column minus that column. And if we do it here, notice I’m moving.
putting I’m putting parentheses around everything just to make sure that because if I don’t do that It loses track and sometimes like in mathematics there There’s like an order of operations that you’re all used to Excel Almost always does what you think it does So I would recommend if you’re going to be putting complicated formulas in into Excel, use a few more parentheses than you expected, just like err on the side of being really explicit with your parentheses and with your multiplication signs. And for complicated stuff, also you notice right here, this would have been a lot to put in one cell. So I created an extra column to put an intermediate step in.
And then we do the same thing, just the same thing here, fill it. Okay, and we’re just out of time. I promised you, I promised, well, I promised the people who emailed me asking for advice about how to do the visualization.
I promised you all that I would show you, but I can’t show you. because of the doubly embarrassing story about my laptop, which won’t run the Excel. So I’m using Excel on the web, and Excel on the web doesn’t allow you to do a 3D scatter plot.
But I’ll show you how to make a 2D scatter plot, and this is almost the same instruction. So we highlight all the data we’re interested in. in.
And we click Insert. And we come over here to this plot ribbon, click the down arrow, and then we’re going to select Scatter Plot. If you have a computer that is younger than mine, mine’s from 2014, so you probably do.
You’ll be able to use Microsoft’s project. on it. And you’ll have an option to pick 3D scatter.
So you’ll select 3D scatter, and it will do almost the same thing that’s going to happen now. Go with one important difference. So you click it.
Palette. Now, what happened here is, we have total cost here. And, oh, so you kind of got confused and…
plotted everything as a function of q1, but what we’ll be able to do in, but that’s because it doesn’t know what to do with that extra variable. The 3D plot will know what to do with that extra variable, and so you’ll end up with something like this. So, q1, q2, q3, q4, q5, q6, q6, q6, q6, q6, q7, q6, q6, q6, q7, q6, q6, q7, q6, q6, q6, q7, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6, q6 so Q So for our two linear terms, it’s just a plane And for our this is like a oh, yeah, you can kind of those scooped right? This is like a piece of paper that’s in bed So this is our one with the squared terms.
And here’s our one that has the interaction. So it’s got a bit of a bend to it. As they both kind of, as they conflict with each other, we have, or sorry, as they conflict with each other, the costs go up in the middle.
So focusing on one or focusing on the other is going to be the way to minimum. costs. Now, when I plotted this– this is the last thing I’ll say about plotting– when I plotted this, I filled in all the spaces.
But all you would need to do in Excel is just do what we did, highlight the data, and ask it to plot. And it’ll just put those points in three dimensions. [VIDEO PLAYBACK] - All right.
On. Tuesday, basically this plotting portion, but kind of simple plotting and then drawing regression lines in there is basically– hopefully, it’s similar to what you saw in counting 2200. If you took it for me, you basically did this next assignment a couple of years ago.
So the next assignment– is going to be less pressure. I wanted to end with the last slide, but since we’re already a minute over, I just want you to know that knowing what to do when you don’t know what to do is not just a skill, it is a superpower. Thank you.
Sorry for keeping you extra in two minutes. Oh, and we’ll have graded the homework within like before the weekend, so today or tomorrow.