Excel Companion Chapter 4
Chapter 4 Index
EC_4 Rates of Change
  Marginal.xls Marginal Analysis
  TimeTemp.xls Average and Instantaneous Rate of Change
  Tangents.xls Slopes of Tangent Lines
  Secants.xls Approximations to the Tangent Line at a Point
  DiffQuot.xls Slope Functions with the Difference Quotient
    Return to Table of Contents
Marginal Analysis Marginal.xls

Rates of change of cost, revenue, and profit functions are often described by economists using the language of marginal analysis. Management decisions within a particular firm or industry often depend on marginal analysis. The general meaning of the term "marginal" in a business context is

"that additional cost, revenue, or profit which results from increasing the quantity by one".

Notation preferred by economists may be slightly different from our mathematical function notation. For example, economists use TC = TVC + TFC to communicate the fact that total cost is the sum of total variable cost plus total fixed cost. They then describe marginal cost as the change in total cost divided by the change in the number of units. This can be represented symbolically as MC = DTC/Dq. . Using function notation we write marginal cost as MC = C(q+1) - C(q) where C is the total cost function. We have already seen linear cost functions which can be represented in function notation as C(q) = v*q + F. Total fixed cost, F, is the intercept on the vertical axis. In the linear case (and only in the linear case) marginal cost, C(q+1) - C(q), is the constant v. We see that marginal cost (variable cost per unit) is independent of the particular level of production for linear cost functions.

For linear revenue functions, total revenue may be defined as R(q) = p*q where the unit price, p, is constant. The marginal revenue of selling one more unit is again independent of q and is simply the slope of the revenue function. Given a pair of linear cost and revenue functions, profit will also be linear and will theoretically continue to increase. Once quantity exceeds the break-even point, the more units the company can make and sell, the higher its profits will be assuming there is continued demand for the product. When we encounter non-linear cost and revenue functions we need more sophisticated mathematical tools to investigate the following important questions:

  1) What are some efficient ways of measuring the rate of change at different levels of production or sales?
  2) Is there a precise way to associate a specific number with the slope of a curve at a point?
  3) Is there a precise way of predicting the quantity at which profit is at a maximum?

Open the file Marginal.xls and go to the worksheet Table.

SITUATION 1: The cost function is a cubic polynomial and the revenue function is linear. Make sure that cells B11, D11, and F11 have the formulas for cost, revenue and profit as shown on the left below in boldface.

  Excel Notation Standard Notation
C(q) =0.002*q^3+22*q+2500 .002q3 + 22q+ 2500
R(q) =80*q 80q
P(q) = R(q) -C(q) =-0.002*q^3+58*q-2500. -.002q3 + 58q - 2500

 

Marginal.xls - Table 1

 Notice that it takes two successive rows to produce one marginal value. For example, the marginal dollar cost, $81.40, of making the 100th unit is the cost of making unit 100 minus the cost of making unit 99. The marginal revenue of making unit 100 is $80.00, which is the slope of the revenue line. The marginal revenue is always $80.00 since the revenue function is linear.

The marginal profit of making and selling the 100th unit is $1300.00-$1301.40 = -$1.40. This number can also be obtained by subtracting marginal cost from marginal revenue, $80.00 - $81.40 = -1.40 at the 100th unit.

Inspect the table above and the graphs below. Observe a steady rise in the marginal cost throughout the entire interval. Marginal cost initially stays below marginal revenue. At these levels, total profit is increasing.

The same scale on the horizontal axis was chosen for the two graphs in order to dramatize the predictive relationships between marginal cost, marginal revenue, and marginal profit and the associated cost, revenue, and profit functions.

Marginal.xls - Graphs

The company is selling additional items for more than the cost of producing them. However, at a production level beyond 98 units, we see that marginal cost surpasses marginal revenue. At this point it costs more to produce the additional item than we get from selling the additional item. The company is still in the black (positive profit), but this cannot continue and eventually profit itself becomes negative. Marginal profit is a way of detecting if profit is headed up (increasing) or down (decreasing). A profit maximizing firm produces and sells an additional unit of output only if marginal revenue is at least as great as marginal cost. For operational decision making, the company is interested in the point at which marginal cost equals marginal revenue.

QUESTIONS:

Enter your name and nine digit ID (with no hyphens) below :

First name Last name ID

1. At some point beyond 140 units, cost actually exceeds revenue and the company begins to lose money (profit turns negative). What is the break-even point near q = 140?
2. Using standard function notation, the marginal profit of making and selling item q+1 can be expressed as where P is profit  
  a) Let q = 39 and find the marginal profit for making and selling the 40th item.
  b) How does the value in a) compare with the difference between the marginal revenue of selling item 40 and the marginal cost of making item 40.
3. a) What do you observe about marginal profit at the point q where marginal revenue and marginal cost have essentially the same value?
  b) What does this tell us about the profit function itself at this value of q?
4. Suppose daily cost and revenue functions are given by

C(q) = q2 + 1500 and R(q) = -q2 + 150q

  
  a) Will the company make a profit if it makes and sells 50 units? yes no
  b) Will the company make a profit if it makes and sells 70 units? yes no
  c) If the company wants to maximize daily profit then how many units should be made and sold?
5. Suppose daily cost and revenue functions are given by

C(q) = 60q + 5600 and R(q) = -2q2 + 280q

  
  a) Will the company make a profit if it makes and sells 80 units?
 yes no
  b) Determine the first break-even point
  c) ) Determine the second break-even point
  d) If the company wants to maximize daily profit then how many units should be made and sold?

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Copyright © Joseph F. Aieta, Babson College 1997