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Study Guides > Business Calculus

Reading: Marginal Cost

Business and Economics Terms

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: [latex-display] MC(q) = C(q + 1) - 1 = \frac {C(q+1)-1}{1} [/latex-display] So this is approximately the same as the derivative of the cost function at q: [latex-display] MC(q) = C \prime(q) [/latex-display] 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.”