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# How to find marginal cost

It’s essential to have a strong understanding of marginal costs if you want to maximize your profits and decrease the cost-per-unit of production. Find out everything you need to know about how to calculate marginal cost.

We’ll explore the marginal cost formula, take you through an example of a marginal cost equation, and explain the importance of marginal costs for business in a little more depth.

## What is marginal cost?

Marginal cost is a term used in economics and accounting that refers to the incremental costs involved in producing additional units. In any marginal cost equation, you’ll need to include the variable costs of production. For example, labor and materials will need to be considered. However, you’ll only need to include the fixed costs of production (i.e., administrative costs, overheads, sales expenses, etc.) if there’s a need to increase your fixed costs to handle the additional output.

## Marginal cost formula

If you want to learn how to calculate marginal cost, you can use the following marginal cost formula:

Marginal Cost = Change in Total Cost / Change in Quantity

Let’s explore the two main elements of the marginal cost formula in a little more depth:

Change in Total Cost – At any level of production, your costs can increase or decrease. If you need to hire an extra worker or purchase more raw materials to make additional units, for example, your production costs will increase. To find out how much your production costs have changed, you can deduct the production cost of batch one from the production cost of batch two.

Change in Quantity – Of course, volumes will also increase or decrease whenever you have differing levels of production. To work out the change to your quantities, you’ll need to deduct the number of goods from your first production run from the number of goods from the second, expanded production run.

## Example of marginal cost

Using the marginal cost formula, let’s explore how marginal cost works in the real world with an example. Imagine that Company A regularly produces 10 handcrafted tables at the cost of £2,000. However, demand spikes and they receive more orders, leading them to purchase more materials and hire more employees. In their next production run, they produce 20 units at the cost of £3,000.

Marginal Cost = (£3,000 – £2,000) / (20 – 10) = £100

In other words, the marginal cost (i.e., the additional expenditure to make another unit) is £100 per table.

## Why is the marginal cost equation important?

Knowing how to calculate marginal costs is vital for a couple of reasons. Most importantly, it provides you with an insight into the efficiency of your production schedule, giving you a way to determine at what point your company will be able to achieve economies of scale (i.e., cost efficiencies resulting in a decreased cost-per-unit). The quicker you can reach an optimum production level, the better for your business. Put simply, if the marginal cost of producing one additional unit is lower than the purchase price, the company can make a profit.

It’s also important to note that your business’s marginal cost curve may begin to increase if your company becomes less productive and suffers from diseconomies of scale (i.e., the inverse of an economy of scale, wherein the business becomes too large, and poor communication, loss of control, and external opposition lead to a rise in per-unit costs). If the amount of revenue you’re generating ( marginal revenue ) is the same – or less – than the marginal cost, you’ll need to call a halt to production, as the cost of production is causing the business to lose money.

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The concept of marginal cost can be difficult for business owners to understand. However, understanding how to calculate marginal cost is essential to good forecasting and business management. With that in mind, we’ve created a step-by-step guide detailing everything from the importance of marginal costs and formula examples.

Below, we’ll examine critical concepts involving the use of marginal cost. Highlights include the relation between variable and fixed costs, the related concept of marginal revenue, and how these are used to determine the optimal point at which a business can make the most profit. In addition, we’ll show you a formula that demonstrates how to find the marginal cost of goods.

## The Importance of Marginal Cost

Knowing how to calculate marginal cost is important because every business should strive to expand to a point where marginal cost is equal to marginal value. So, let’s start by exploring what marginal cost is and how to find the marginal costs associated with your business.

Marginal cost is the added cost to produce an additional good. For example, say that to make 100 car tires, it costs $100. To make one more tire would cost$80. This is then the marginal cost: how much it costs to create one additional unit of a good or service.

The costs of production determine the marginal cost. These include fixed and variable costs. Fixed costs are things like monthly rent and utilities. Variable costs are things that can change over time, such as costs for labor and raw materials.

A good example is if demand for running shoes for a footwear company increases more machinery may be needed to expand production and is a one-off expense. However, it does need to be accounted for at the point the purchase takes place. Usually, marginal costs include all costs that vary with increases in production. A U-shaped short-run Average Cost (AC) curve. AVC is the Average Variable Cost, AFC the Average Fixed Cost, and MC the marginal cost curve crossing the minimum of both the Average Variable Cost curve and the Average Cost curve.

## The Marginal Cost Curve

The marginal cost curve is the relation of the change between the marginal cost of producing a run of a product, and the amount of the product produced. In classical economics, the marginal cost of production is expected to increase until there is a point where producing more units would increase the per-unit production cost. Calculating marginal cost and understanding its curve is essential to determine if a business activity is profitable.

## The Marginal Cost Formula

How do you find the marginal cost? There’s a mathematical formula that expresses the change in the total cost of a good or product that comes from one additional unit of that product. Knowing this formula is essential in learning how to calculate marginal cost. It is called the marginal cost equation or marginal cost formula.

Marginal Cost = (Changes in Costs)/(Changes in Quantity)

This is an important formula for cost projections and determining whether or not a business activity is profitable.

• Change in Total Cost is the usual net fixed and variable costs that go into the production of goods. Variable costs include labor, raw materials, and so on.
• Change in Quantity refers to an obvious increase in the number of goods produced.

## A Marginal Cost Formula Example

Here is an example of how to calculate marginal cost:

Big Dynamo is a toy company that produces robot toys. Every month, they produce 2,000 robot toys for a total cost of $200,000. They expect to produce 4,000 robot toys next month for$250,000.

This marginal cost calculator helps you calculate the cost of an additional units produced. Marginal cost is the change in cost caused by the additional input required to produce the next unit. It may vary with the number of products provided by the company. Based on this value, it may be easier to decide if production should increase or decrease. You may find a marginal cost calculator under different names, such as an incremental cost calculator or a differential cost calculator, but they are all related to the same topic. However, marginal cost is not the same as margin cost! In this article, you can find more details on how to calculate the marginal cost, and the marginal cost formula behind it.

## How to calculate the marginal cost

The steps below will help you understand how to calculate the marginal cost:

1. Find out how much your costs will increase once you produce any additional units;
2. Think about how many additional products you would like to create;
3. Divide the additional cost from point 1 by the extra units from point 2; and
4. Thats it, you have calculated the marginal cost!

Below you may find the marginal cost formula if you prefer a mathematical approach.

## Marginal cost formula

The formula for the marginal cost is quite simple:

MC – marginal cost;

ΔTC – change in the total cost; and

ΔQ – change in the total quantity.

For example, imagine that your company produces chairs. Every month there are new 10,000 chairs created, which costs the company a total of $5,000. You may wonder how much it would cost to produce an additional 2,000 chairs, and, if so, you should use the marginal cost calculator. If 12,000 chairs costs$5,500, input this data into the marginal cost formula from above:

MC = ΔTC/ΔQ = (5,500 – 5,000) / (12,000 – 10,000) = 500 / 2,000 = $0.25 What the tells us is that it costs your company$0.25 to produce chair number 12,000. You may wonder why this final chair costs less than than the cost per unit for 10,000 chairs. To understand this, you should learn more about economies of scale.

## Economies of scale

As you increase the number of units produced, you may find that the cost per unit decreases. This is because it is cheaper to create the next unit – our marginal cost, as your fixed costs remain unchanged. For example, you do not have to pay more for your warehouse if you produce one more unit of the product (unless it is more than your warehouse’s capacity). Your additional cost of producing one extra product depends mostly on the value of the product itself – materials, workers wages, etc. Because of that, your marginal cost may decrease.

Using this calculator will help you calculate the cost of the next unit, and decide if it is worth it to increase production. Once you choose to change your output, you may find it encouraging to calculate your new potential profit!

You may wonder if increasing production is always profitable. Well, that depends on your capacity. Sometimes you may incur additional costs, like a new production machine as the one you currently have is not able to produce any more product over a specific period. You may find it useful to read the next section to understand how to find the most profitable quantity to produce.

## How many units should I produce?

Knowing how to calculate the marginal cost is the first step towards finding the best quantity to produce. The second step is to consider marginal revenue. This value is calculated similarly to marginal cost, but, instead of additional cost, it uses the additional revenue the extra unit produced, ΔTR:

To find the perfect quantity, you have to find the value for which marginal cost, MC will be equal to marginal revenue:

You can think about it in another way – for any change in quantity, the new marginal cost and marginal revenue would be the same, so it is enough to compare the change in total cost and the change in total revenue:

Say that you have a cost function that gives you the total cost, C(x), of producing x items (shown in the figure below). The derivative of C(x) at the point of tangency gives you the slope of the tangent line. Slope equals rise/run, right? So when the run equals 1, the rise equals the slope (which equals the derivative). On the little triangle under the tangent line, you run across 1 and then you rise up an amount called the marginal cost. And thus the derivative equals the marginal cost, get it?

Going 1 to the right along the curving cost function itself shows you the exact increase in cost of producing one more item. If you look very closely at the right side of the above figure, you can see that the extra cost goes up to the curve, but that the marginal cost goes up a tiny amount more to the tangent line, and thus the marginal cost is a wee bit more than the extra cost (if the cost function happened to be concave up instead of concave down like it is here, the marginal cost would be a tiny bit less than the extra cost).

So, because the tangent line is a good approximation of the cost function, the derivative of C — called the marginal cost — is the approximate increase in cost of producing one more item. Marginal revenue and marginal profit work the same way.

Before doing an example involving marginals, there’s one more piece of business to take care of. A demand function tells you how many items will be purchased (what the demand will be) given the price. The lower the price, of course, the higher the demand.

You might think that the number purchased should be a function of the price — input a price and find out how many items people will buy at that price — but traditionally, a demand function is done the other way around. The price is given as a function of the number demanded. That may seem a bit odd, but the function works either way. Think of it like this: If a retailer wants to sell a given number of items, the demand function tells them what the selling price should be.

Okay, so here’s the example. A widget manufacturer determines that the demand function for their widgets is

where x is the demand for widgets at a given price, p. The cost of producing x widgets is given by the following cost function:

Determine the marginal cost, marginal revenue, and marginal profit at x = 100 widgets.

## Marginal cost ## Marginal profit

Marginal profit is the derivative of the profit function, so take the derivative of P(x) and evaluate it at x = 100. So, selling the 101st widget brings in an approximate profit of $35. By the way, while the above math is exactly what you’d want to do if you were asked only to compute the marginal profit, did you notice that it was unnecessary in this example? Once you know the marginal cost and the marginal revenue, you can get marginal profit with the following simple formula: Just about everything you do in your business comes with a cost. Whether it’s time, money, effort, or something else, you pay a price. But, what happens when you set a limit for production and have to produce more than your set limit? You encounter what’s known as marginal cost. What is marginal cost? ## Marginal cost definition The marginal cost meaning is the expense you pay to produce another service or product unit beyond what you intended to produce. So if you planned to produce 10 units of your product, the cost to produce unit 11 is the marginal cost. Businesses typically use the marginal cost of production to determine the optimum production level. Once your business meets a certain production level, the benefit of making each additional unit (and the revenue the item earns) brings down the overall cost of producing the product line. Marginal costs include more than just the cost of materials. The marginal cost of production includes everything that varies with the increased level of production. For example, if you need to rent or purchase a larger warehouse, how much you spend to do so is a marginal cost. Marginal cost is not the same as the markup on your products. Markup is how much more your selling price is than the amount the item costs you to produce. Ponder no more! Download our FREE guide, Price to Sell and Profit, for the scoop. ## How to calculate marginal cost (marginal cost formula) Before we dive into the marginal cost formula, you need to know what costs to include. Marginal costs include variable and fixed costs. Variable costs include the labor and materials that go into your final product’s production. Fixed costs include expenses like administrative work and overhead. Fixed costs do not change if you increase or decrease production levels. So, you can spread the fixed costs across more units when you increase production (and we’ll get to that later). Now that you know the difference between the types of costs, let’s look at the marginal cost formula and how to find marginal cost. Your marginal costs is the total change in costs divided by the change in quantity: Marginal Cost = Change in Costs / Change in Quantity ### Change in costs So, what is the change in costs you need for the marginal cost equation? Each production level may see an increase or decrease during a set period of time. This can occur when you need to produce more or less volume. An increase or decrease in production costs during a set period of time is a change in costs. To calculate the total change in costs, subtract the previous production’s costs from the current batch’s costs: Change in Costs = Production Run B Costs – Production Run A Costs ### Change in quantity The change in quantity is the difference between how many units your business produces between production runs. A change in quantity can be an increase or decrease. To determine the change in quantity, subtract the number of units your business produced in the first production run from the number of units in the second production run: Change in Quantity = Production Run B Unit Amount – Production Run A Unit Amount ## Marginal cost examples Before we look at some marginal cost examples, let’s find out the cost of production for a typical business. Your business produces screen-printed T-shirts. Each T-shirt you produce requires$5.00 of T-shirt and screen printing materials to produce, which are your variable costs. You spend $2,000 each month on fixed costs (e.g., overhead). You produce 500 T-shirts each month. To find how much it costs to produce each T-shirt on a standard production run, divide the fixed costs by the unit amount and add the variable costs: Cost Per Unit = ($2,000 / 500) + $5.00 Cost Per Unit =$9.00

Your standard costs are $9.00 per unit. To find your total cost of production, multiply the cost per unit by the number of units: Cost of Production =$9.00 X 500

Cost of Production = $4,500 Your total cost of production is$4,500 per month for 500 T-shirts.

### Example 1

You decide to increase your T-shirt production and produce 750 T-shirts per month. Your variable and fixed costs do not change. To find the new production cost, divide the new unit amount by the fixed costs and add the variable cost:

Cost Per Unit = ($2,000 / 750) +$5.00

Cost Per Unit = $7.67 Find your change in costs by multiplying the cost per unit by the unit amount. Subtract the new cost of production from your old cost: Change in Costs = ($7.67 X 750) – $4,500 Change in Costs =$1,252.50

Your change in costs is $1,252.50. To find your change in quantity, subtract your original production unit amount from the new production amount: Change in Quantity = 750 – 500 Change in Quantity = 250 You produce 250 more units per month. After you determine the change in costs and the change in quantity, calculate the marginal cost of production: Marginal Cost =$1,252.50 / 250

Marginal Cost = $5.01 Your marginal cost of production is$5.01 per unit for every unit over 500. In this example, it costs $0.01 more per unit to produce over 500 units. ### Example 2 In this example, you continue to produce 750 T-shirts but purchase a new facility. The new facility increases your fixed costs by$200 per month. Your new fixed costs are $2,200 ($2,000 + $200). Calculate your new costs per unit: Cost Per Unit = ($2,200 / 750) + $5.00 Cost Per Unit =$7.93

Your new cost per unit is $7.93. Calculate the new change in costs: Change in Costs = ($7.93 X 750) – $4,500 Change in Costs =$1,447.50

Because your quantity did not change, you can use the marginal cost formula to calculate the new marginal cost of production:

Marginal Cost = $1,447.50 / 250 Marginal Cost =$5.79

Your marginal cost pricing is $5.79 per additional unit over the original 500 units. In this example, you can see it costs$0.79 more per unit over the original 500 units you produced ($5.79 –$5.00).

## Why are marginal costs important?

The marginal cost of production helps you find the ideal production level for your business. You can also use it to find the balance between how fast you should produce and how much production is too low to help growth.

And by figuring out your marginal cost, you can more accurately determine your margin vs. markup to better price your products and turn a profit. Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

• Explanation
• Transcript

The marginal cost function is the derivative of the total cost function, C(x). To find the marginal cost, derive the total cost function to find C'(x). This can also be written as dC/dx — this form allows you to see that the units of cost per item more clearly. So, marginal cost is the cost of producing a certain numbered item.

Let’s do a problem that involves marginal cost. I specifically want to find out how marginal cost actually compares to the cost of producing one more item. Let’s take a look at our skateboard example. Suppose C(x) is the total cost of producing x skateboards. This is our cost function; C(x) is 1800 plus 10x plus 0.02x². Of course the cost is going to be in dollars.

We’ll do three things. We’ll find the marginal cost function, that’s just C'(x). B; we’ll find the c'(500) and give the units. In part c, we’ll find the actual cost of producing the 501st skateboard, and compare that with our answer top part b.

We want to see really how good of an approximation the marginal cost is for producing that 501st skateboard. So first part a; find the marginal cost function. Most important thing to remember about marginal cost is it’s just the derivative of cot. So the marginal cost is going to be C'(x). That’s going to be well the derivative of 1800 is 0, the derivative of 10x is 10 plus, the derivative of 0.02x² is 2 times 0.02, 0.04x. That’s pretty easy. So this is my marginal cost function.

Part b; find the marginal cost at 500, and give units. So I’m just going to plug 500 into this function. C'(500) is 10 plus 0.04 times 500. Now 0.04 times 500, whenever I’m multiplying by decimals, I can think of this as multiplying by 4 then dividing by 100. Multiplying by 4 gives me 2,000. Dividing by a 100 gives me 20. 20 and 10 is 30. So that’s 30, and what are the units?

Let’s remember that C'(500) is actually the same as dc/dx. So I can write c'(x) this way. When you write the derivative this form, it’s much easier to see what the units would be. Units of the cost function divided by units of x. The cost function has units of dollars. X is just numbers of skateboards, so this would be dollars per skateboard, and that’s what we have here; dollars per skateboard. So that’s a nice way to get the units for a derivative was to look at it form.

In part c, we want to find the actual cost of the 501st. Let me just sketch out what I’m going to do here. The actual cost is going to be C(501) minus c(500). Let’s see that this is a much more complicated computation than what we just did, but it will give us the actual cost of the 501st skateboard. So let’s take this calculation over here to the right.

So I need C(501) minus C(500). Let me calculate each of those separately. First C(501). This is my cost function. It’s 1800 plus 10 times 501 plus 0.02 501². So that’s 1800 plus 10 times 501 is 5,010 plus 0.02 times 501² is 251,001. Then I’ve got to multiply this by 0.02. That’s the same as multiplying by 2, and dividing by a 100. Multiplying by 2 would give me 502,002. Dividing by a 100 would give me that. So plus 5,010 plus 1800. Now adding all this together, I notice I have 10,000, 30, and 2 cents. Plus 1800 is 11,832, and 2 cents. That’s my cost at 501 skateboards.

What’s my cost at 500? I have to use this function again 1,800 plus 10 times 500 plus 0.02 times 500². That’s just 1,800 plus 5,000 plus 500² is 250,000 times 0.02 again multiply by 2,500,000, and divide by a 100 means I put a decimal point right there. So this is 5,000 plus another 5,000 plus 1800. This is going to give me 11,800.

Now the difference C(501) minus C(500) is going to be $30, and 2 cents. Now this is the actual cost of producing the 501st skateboard. Look at all the work I just did just to find that the actual cost is$30, and 2 cents. That’s the actual cost of that 501st skateboard.

My approximation using marginal cost over here was $30 per skateboard. This was a lot easier to calculate too. So this is the value of marginal cost. Take the derivative, plug in 500, and you get a very accurate approximation of the cost of one more skateboard, versus this calculation did over here which took me half of the board. So marginal cost is a really valuable concept. It gets you a very quick estimate too of the cost of producing one more skateboard. “Economics is the painful elaboration of the obvious” – Anonymous. That quote might seem quite relevant when the biggest conclusion of our last section was that you should do something if the benefits outweigh the costs. While sometimes economics can seem obvious, it is important to first understand how a rational consumer should behave before seeing how we fail to meet that standard. ## Marginal Analysis In the last section we showed how to make a binary decision, but not all decisions fit that category. Many are ‘how much’ decisions. For example, if you have decided to go clubbing, how many drinks do you buy? This is a decision where we use marginal analysis. Marginal analysis is the process of breaking down a decision into a series of ‘yes or no’ decisions. More formally, it is an examination of the additional benefits of an activity compared to the additional costs incurred by that same activity. To make a decision using marginal analysis, we need to know the willingness to pay for each level of the activity. As mentioned, this is also known as the marginal benefit from an action. To decide how many drinks to buy, you have to make a series of yes or no decisions on whether to buy an additional drink. In Table 1.3a the marginal benefit is diminishing. This means that you are willing to pay more for the 1st drink than the next. Your friends are all drinking, so you are likely willing to pay quite a lot for your 1st drink. By the 4th, you may feel as though you do not need another. So how many drinks will you buy if the cost is$7? To make this decision, we must use marginal analysis for each level. This means comparing our marginal benefit with marginal cost of an additional unit of activity. In this case marginal cost is just equal to $7. For the 1st Drink: MB =$20 > MC = $7, you should buy the drink. For the 2nd Drink: MB =$12 > MC = $7, you should buy the drink. For the 3rd Drink: MB =$6 < MC = $7, you should not buy the drink. With this simple process we can easily see that you will buy 2 drinks, unless there is a price change. ## Net Benefit What is our net benefit from the actions, or how much ‘happiness’ have we gained? To calculate, all we have to do is add up our benefits and subtract our costs. Total Benefit =$20 + $12 =$32

Total Cost = $7 +$7 = $14 Net Benefit =$32 – $14 =$18

It is important to recognize that our act of marginal analysis has maximized this benefit. Consider what would happen if we purchased 3 drinks.

Total Benefit = $20 +$12 + $6 =$38

Total Cost = $7 +$7 + $7 =$21

Net Benefit = $38 –$21 = $17 Note that although total benefit is more than it was previously, net benefit is lower. Looking closer we can see that net benefit fell because our total costs rose ($14 –> $21) by more than our total benefits ($32 –> $38). As a quick rule: When total benefits rise more than total costs, then the action is logical. When total costs rise more than total benefits, then the action is illogical. This is why we look at the marginal net benefit of a decision, rather that the total. It is as though all the previous actions are ‘sunk’. We can calculate the marginal net benefit of a decision by subtracting marginal cost from marginal benefit. Marginal net benefit of the first drink is$13 ($20 –$7), the 2nd is $5 ($12 – $7), and the third is -$1 ($6 –$7). As long as the marginal net benefit is positive, we should increase our activity!

## Summary

Marginal analysis is an essential concept for everything we learn in economics, because it lies at the core of why we make decisions. We have just scratched the surface of it now, but will go more in depth in Topic 3. For now, we will turn our attention to a slightly different topic – trade.

## Glossary

Marginal Analysis The examination of the additional benefits of an activity compared to the additional costs incurred by that same activity

Marginal Benefit

The additional satisfaction one gains from an additional unit of an activity

Marginal Cost

The additional costs from an additional unit of an activity Marginal Net Benefit The difference between the marginal benefits and marginal costs of an action

### Exercises 1.3

1. According to marginal analysis, optimal decision-making involves:

a) Taking actions whenever the marginal benefit is positive.
b) Taking actions only if the marginal cost is zero.
c) Taking actions whenever the marginal benefit exceeds the marginal cost.
d) All of the above.

2. Jane’s marginal benefit per day from drinking coke is given in the table below. This shows that she values the first coke she drinks at $1.20, the second at$1.15, and so on.
If the price of coke is \$1.00, the optimal number of cokes that Jane should drink is: a) 1.
b) 2.
c) 3.
d) 4.

Suppose you are producing and selling some item. The profit you make is the amount of money you take in minus what you have to pay to produce the items. Both of these quantities depend on how many you make and sell. (So we have functions here.) Here is a list of definitions for some of the terminology, together with their meaning in algebraic terms and in graphical terms.

• Your cost is the money you have to spend to produce your items.
• The Fixed Cost (FC) is the amount of money you have to spend regardless of how many items you produce. FC can include things like rent, purchase costs of machinery, and salaries for office staff. You have to pay the fixed costs even if you don’t produce anything.
• The Total Variable Cost (TVC) for q items is the amount of money you spend to actually produce them. TVC includes things like the materials you use, the electricity to run the machinery, gasoline for your delivery vans, maybe the wages of your production workers. These costs will vary according to how many items you produce.
• The Total Cost (TC) for q items is the total cost of producing them. It’s the sum of the fixed cost and the total variable cost for producing q items.
• The Average Cost (AC) for q items is the total cost divided by q, or TC/q. You can also talk about the average fixed cost, FC/q, or the average variable cost, TVC/q.
• The Marginal Cost (MC) at q items is the cost of producing the next item. Really, it’s MC(q) = TC(q + 1) – TC(q).
In many cases, though, it’s easier to approximate this difference using calculus (see Example below). And some sources define the marginal cost directly as the derivative, MC(q) = TC′(q).
In this course, we will use both of these definitions as if they were interchangeable.
• Demand is the functional relationship between the price p and the quantity q that can be sold (that is demanded). Depending on your situation, you might think of p as a function of q, or of q as a function of p.
• Your revenue is the amount of money you actually take in from selling your products. Revenue is price × quantity.
• The Total Revenue (TR) for q items is the total amount of money you take in for selling q items.
• The Average Revenue (AR) for q items is the total revenue divided by q, or TR/q.
• The Profit (π) for q items is TR(q) – TC(q).
• The average profit for q items is π/q. The marginal profit at q items is π(q + 1) – π(q), or π′(q)

### Example

Why is it OK that are there two definitions for Marginal Cost (and Marginal Revenue, and Marginal Profit)?

We have been using slopes of secant lines over tiny intervals to approximate derivatives. In this example, we’ll turn that around—we’ll use the derivative to approximate the slope of the secant line.

Notice that the “cost of the next item” definition is actually the slope of a secant line, over an interval of 1 unit:

$MC(q) = C(q + 1) – 1 = \frac <1>$

So this is approximately the same as the derivative of the cost function at q:

$MC(q) = C \prime(q)$

In practice, these two numbers are so close that there’s no practical reason to make a distinction. For our purposes, the marginal cost is the derivative is the cost of the next item.

Graphical Interpretations of the Basic Business Math Terms

Illustration/Example:

Here are the graphs of TR and TC for producing and selling a certain item. The horizontal axis is the number of items, in thousands. The vertical axis is the number of dollars, also in thousands.

First, notice how to find the fixed cost and variable cost from the graph here. FC is the y-intercept of the TC graph. (FC = TC(0).) The graph of TVC would have the same shape as the graph of TC, shifted down. (TVC = TC – FC.)

We already know that we can find average rates of change by finding slopes of secant lines. AC, AR, MC, and MR are all rates of change, and we can find them with slopes, too.

AC(q) is the slope of a diagonal line, from (0, 0) to (q, TC(q)). AR(q) is the slope of the line from (0, 0) to (q, TR(q)).

MC(q) = TC(q + 1) – TC(q), but that’s impossible to read on this graph. How could you distinguish between TC(4022) and TC(4023)? On this graph, that interval is too small to see, and our best guess at the secant line is actually the tangent line to the TC curve at that point. (This is the reason we want to have the derivative definition handy.)

MC(q) is the slope of the tangent line to the TC curve at (q, TC(q)). In a similar way, MR(q) is the slope of the tangent line to the TR curve at (q, TR(q)).

Profit is the distance between the TR and TC curve. If you experiment with your clear plastic ruler, you’ll see that the biggest profit occurs exactly when the tangent lines to the TR and TC curves are parallel. This is the rule “profit is maximized when MR = MC.”

A very clear way to see how calculus helps us interpret economic information and relationships is to compare total, average, and marginal functions.

Take, for example, a total cost function, TC:

For a given value of Q, say Q=10, we can interpret this function as telling us that: when we produce 10 units of this good, the total cost is 190. We would like to learn more about how costs evolve over the production cycle, so let’s calculate average cost, which is total cost divided by the number of units produced, or Q:

Therefore, when we produce 10 units of this good, the average cost per unit is 19. This is somewhat deceptive, however, because we still don’t know how costs evolve or change as we produce. For example, the first unit (Q = 1) cost 10 to produce. Obviously, if the average ends up being 19, and the first unit cost 10, then the cost of producing a unit must be changing as we produce different units. Alternatively, to be more technical, the change in total cost is not the same every time we change Q. Let’s define this change in total cost for a given change in Q as the marginal cost.

Sound familiar? The slope is defined as the rate of change in the Y variable (total cost, in this case) for a given change in the X variable (Q, or units of the good). Therefore, taking the first derivative, or calculating the formula for the slope can determine the marginal cost for a particular good.

What about the change in marginal cost? That way, we can not only evaluate costs at a particular level, but we can see how our marginal costs are changing as we increase or decrease our level of production. Thanks to our calculus background, it’s clear that the change in marginal cost or change in slope can be calculated by taking the second derivative.

These three equations now give us a considerable amount of information regarding the cost process, in a very clear format. For example, calculate the marginal cost of producing the 100th unit of this good.

Now, suppose your boss wants you to forecast costs for the 101st unit. You can recalculate marginal cost, or you can note that the second derivative tells you that the marginal cost is expected to change by an increase of two, for every one unit increase in Q. Therefore,

To sum up, you can start with a function, take the first and second derivatives and have a great deal of information concerning the relationship between the variables, including total values, changes in total values, and changes in marginal values.

## Characteristics of relative and absolute maxima and minima

The first and second derivatives can also be used to look for maximum and minimum points of a function. For example, economic goals could include maximizing profit, minimizing cost, or maximizing utility, among others.

In order to understand the characteristics of optimum points, start with characteristics of the function itself. A function, at a given point, is defined as concave if the function lies below the tangent line near that point. To clarify, imagine a graph of a parabola that opens downward. Now, consider the point at the very top of the parabola. By definition, a line tangent to that point would be a horizontal line.

It’s clear that the graph of the top section of the parabola, in the neighborhood of the point, all lies below the tangent line, therefore, the graph is concave in the neighborhood of that point.

Note how much care is being taken to limit the discussion of concavity to the part of the function near the point being considered. Suppose the function is a higher order polynomial, one that takes the shape of a curve with 2 or more turning points. It would be easy to imagine a function where part was below the horizontal tangent line, turned again, and came back up past the line. The definition of concavity refers only to the part of the function near the point where the tangent line touches the curve, it isn’t required to hold everywhere on the curve.

Consider the tangent line itself. Recall from past section on linear functions that the slope of a horizontal line or function is equal to zero. Therefore, the slope at the top or turning point of this concave function must be zero. Another way to see this is to consider the graph to the left of the turning point. Note that the function is upward-sloping, ie has a slope greater than zero. The section of the graph to the right of the turning point is downward-sloping, and has negative slope, or a slope less than zero.

As you look at the graph from left to right, you can see that the slope is first positive, becomes a smaller positive number the closer you get to the turning point, is negative to the right of the turning point, and becomes a larger negative number the further you travel from the turning point. Since this is a continuous function, there must be a point where the slope crosses from positive to negative. In other words, for an instant, the slope must be zero. This point we have already identified as the turning-point.

There is a much easier way to identify what’s going on, however. Recall that second derivatives give information about the change of slope. We can use that in conjunction with the first derivative at increasing points of x (as you travel left to right on the graph) to determine identifying characteristics of functions.