CHAPTER TWO
OPTIMAL DECISIONS USING
MARGINAL ANALYSIS
OBJECTIVES
1. To introduce the basic economic model of the firm (A Simple Model of
the Firm)
– The main focus is on determining the firm’s profit-maximizing level
of output.
– The main assumption is that there is a single product (or multiple,
independent products) with deterministic demand and cost.
2. To depict the behavior of price, revenue, cost, and profit as output
varies. (A Microchip Manufacturer)
3. To explain the notion of marginal profit (including its relationship to
calculus) and show that maximum profit occurs at an output such that
marginal profit equal zero. (Marginal Analysis)
4. To reinterpret the optimality condition in terms of the basic
components, marginal revenue and marginal cost. (Marginal Revenue
and Marginal Cost)
5. To illustrate the uses of sensitivity analysis. (Sensitivity Analysis)
TEACHING SUGGESTIONS
I. Introduction and Motivation
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A. This is a “nuts and bolts” chapter. Because it appears up front in the
text, it’s important to explain the motivation and assumptions. It is a
good idea to remind students of the following points.
1) The model of the firm is deliberately simplified so that its logic is laid
bare. Many additional complications will be supplied in later chapters.
The key simplifications for now are:
• The model is of a generic firm. Although microchips are chosen to
make the discussion concrete, there is no description of the kind of
market or the nature of competition within it. The description and
analysis of different market structures comes in Chapters 7 through 10.
• Profit is the sole goal of the firm; price and output are the sole decision
variables.
• The description of demand and cost is as “bare bones” as it gets. The
demand curve and cost function are taken as given. (How the firm
might estimate these are studied in Chapters 3 through 6.)
B. In general, our policy is to use extended decision examples, different
than the ones in the text, to illustrate the most important concepts.
(Going over the same examples pushes the boredom envelope.) In the
present chapter, we make an exception to this rule. It is important to
make sure that students with different economic and quantitative
backgrounds all get off roughly on the same foot. Reviewing a familiar
example (microchips) makes this much easier.
II. Teaching the “Nuts and Bolts”
A. Graphic Overview. The text presents the revenue, cost, and profit
functions in three equivalent forms: in tables, in graphs, and in
equations. In our view, the best way to convey the logic of the
relationships is via graphs. (The student who craves actual numbers
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can get plenty of them in the text tables.) Here is one strategy for
teaching the nuts and bolts:
1. Using the microchip example, depict the demand curve, briefly note its
properties and demand equation (in both forms).
2. Next focus on revenue, noting the tradeoff between price and quantity.
Present and justify the revenue equation. Graph it and note its
properties.
3. Repeat the same process with the cost function (reminding students
about fixed versus variable cost). At this point, your blackboard graph
should be a copy of Figure 2.8. Steps 1-3 should take no more than 20
minutes.
4. Since the gap between the revenue and cost curves measures profit, one
could find the optimal output by carefully measuring the maximum gap
(perhaps using calipers). Emphasize that marginal analysis provides a
much easier and more insightful approach. Point out the economic
meaning of marginal cost and marginal revenue. Note that they are the
slopes of the respective curves.
5. Next argue that the profit gap increases (with additional output) when
MR > MC but narrows when MR < MC. (On the graph, select
quantities that are too great or too small to make the point.) Identify Q*
where the tangent to the revenue curve is parallel to the slope of the
cost function. In short, optimal output occurs where MR = MC.
B. Other Topics. The approach in part A provides a simple way of
conveying the basic logic of marginal analysis using the components of
MR and MC. Once this ground is covered, the instructor should
emphasize other basic points:
1. The equivalence between Mπ = 0 and MR = MC.
2. Calculus derivations of Mπ, MR, and MC.
3. The exact numerical solution for the microchip example.
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4. The graphs of MR and MC and an exploration of comparative statics
effects (shifts in the curves) and the effects on Q*.
C. Applications. Besides the applications in the text, the following
problems are recommended: Problem 1 (a quick but important check),
Problems 6, 7 and 9 (numerical applications), Problem 11 (the general
solution), and Problem 12. (If the class has a good grasp of this last
problem, nothing else will seem difficult.)
Here is a stylized example that nicely illustrates marginal analysis.
1. SITING A SHOPPING MALL
A real-estate developer is planning the construction of a large
shopping mall in a coastal county. The question is where to locate it.
To help her in the decision, the developer has gathered a wealth of
information, including the stylized “map” of the region in the
accompanying figure. The county’s population centers run from west
to east along the coast (these are labeled A to H), with the ocean to the
north. Since available land and permits are not a problem, the
developer judges that she can locate the mall anywhere along the
coast, that is, anywhere along line segment AH. In fact, the mall
would be welcome in any of the towns due to its potential positive
impact on the local economy.
According to an old adage, “The three most important factors in
the real-estate business are location, location, and location.”
Accordingly, the developer seeks a site that is proximate to as many
potential customers as possible. A natural measure of locational
convenience is the total travel miles (TTM) between the mall and its
customer population. Thus, Figure 2.1 notes the distances between
towns in the county. It also shows the potential number of customers
per week in each town. The developer’s key question is: Where along
the coast should the mall be located to minimize the total travel miles?
Figure 1 Locating a Shopping Mall
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Number of Customers per week (thousands)
15 10 10 10 5 20 10 15
West ______________________________________ East
A| B C D E F G H
3.0 3.5 2.0 2.5 4.5 2.0 4.5
Distance between Towns (miles)
Answer
We could try to find the best location by brute force – by selecting
alternative sites and computing the TTM for each one. For example,
the TTM at the possible site labeled X (1 mile west of town C) is
(5.5)(15) + (2.5)(10) + (1.0)(10) + (3.0)(10)
+ (5.5)(5) + (10.0)(20) + (12.0)(10) + (16.5)(15) = 742.5.
The TTM is found by multiplying the distance to the mall by the
number of trips for each town (beginning with A and ending with H)
and summing. However, the method requires a good deal of
computational effort while offering no guarantee that an optimal
location (i.e., one that has the lowest TTM of all possible candidates)
will be found. The method only claims that its choice is the best of the
limited number of candidates for which TTMs have been computed.
It is far easier to determine the mall’s optimal location using
marginal analysis. Hint: Begin with an arbitrary location, say, point X.
(Do not compute its TTM.) Instead, consider a small move to a nearby
site, such as town C. Then compute the change in the TTM of such a
move.
Marginal analysis identifies the optimal location as town E. The
demonstration involves a number of steps following a very simple
logic. First consider the move from location X to Town C. Making
this move, we see that TTM must decline. The eastward move means
a 1-mile reduction in travel distance for all customers at C or farther
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east (70,000 trip-miles in all). Therefore, the TTM is reduced by this
amount. Of course, travel distances have increased for travelers at or
to the west of X. For these customers, the TTM increase is 25,000
trip-miles. Therefore, the net overall change in TTM is -70,000 +
25,000 = -45,000 trip-miles. Total TTM has declined because the site
moved toward a greater number of travelers than it moved away from.
Town C, therefore, is a better location than site X.
Because the original move was beneficial, we try moving farther
east, say, to town D. Again, the move reduces the TTM. (Check this.)
What about a move east again to town E? This brings a further
reduction. What about a move to town F? Now we find that the TTM
has increased. (By how much?) Moreover, any further moves east
would continue to increase the TTM. Thus, town E is the best site.
The subtlety of the method lies in its focus on changes. One need
never actually calculate a TTM (or even know the distances between
towns) to prove that town E is the optimal location. (We can check
that town E’s TTM is 635.) One requires only some simple reasoning
about the effects of changes.
Here is a second application.
2. a. For five years, an oil drilling company has profitably operated in the
state of Alaska (the only place it operates). Last year, the state
legislature instituted a flat annual tax of $100,000 on any company
extracting oil (or natural gas) in Alaska. How would this tax affect the
amount of oil the company extracts? Explain.
b. Suppose instead that the state imposes a well-head tax, let’s say a tax of
$10.00 on each barrel of oil extracted. Answer the questions of part a.
c. Finally, suppose that the state levies a proportional income tax (say
10% of net income). Answer the questions of part a. What would be
the effect of a progressive tax?
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d. Now suppose that the company has a limited number of drilling rigs
extracting oil at Alaskan sites and at other sites in the United States.
What would be the effect on the company’s oil output in Alaska if the
state levied a proportional income tax as in part c?
Answer
a. This tax acts as a fixed cost. As long as it remains profitable to produce
in Alaska, the tax has no effect on the firm’s optimal output.
b. The well-head tax increases the marginal cost of extraction by $10.00
per barrel. The upward shift in MC means the new intersection of MR
and MC occurs at a lower optimal level of output.
c. The income tax (either proportional or progressive) has no effect on the
company’s optimal output. For instance, suppose that the company’s
after-tax income is: = .9(R-C) under a 10% proportional tax. To
maximize its after-tax income, the best the company can do is to
continue to maximize its before-tax income. Another way of seeing this
is to note that the tax causes a 10% downward shift in the firm’s MR
and MC curves. With the matching shift, the new intersection of MR
and MC is at the same optimal quantity as the old intersection.
d. When the firm operates in multiple states with limited drilling rigs,
using a rig in Alaska means less oil is pumped (and lower profit is
earned) somewhere else. There is an opportunity cost to Alaskan
drilling. Thus, one can argue that before the tax, the company should
have allocated rigs so as to equate marginal profits in the different
states. With the tax, the marginal profit in Alaska is reduced, prompting
the possible switch of rigs from Alaska to other (higher marginal profit)
locations.
D. Mini-case: Apple Computer in the Mid-1990s
The mini-case reproduced on the next page provides a hands-on
application of profit maximization and marginal analysis.
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Answer
a. Clearly, the period 1994-1995 was marked by a significant adverse
shift in demand against Apple due to major enhancements of
competing computers: lower prices, better interfaces (Windows),
sales to order (Dell), and more abundant software.
b. Setting MR = MC implies 4,500 – .3Q = 1,500, so Q* = 10,000 units
and P = $3,000. Given 1994’s state of demand, Apple’s 1994
production strategy was indeed optimal.
c. In 1995, demand and MR have declined significantly. Now, setting
MR = MC implies 3,900 – .3Q = 1,350, so Q* = 8,500 units and P =
$2,625. Apple should cut its price and its planned output.
Apple Computer in the Mid 90s
Between 1991 and 1994, Apple Computer engaged in a holding action in the desktop
market dominated by PCs using Intel chips and running Microsoft’s operating system.1
In 1994, Apple’s flagship model, the Power Mac, sold roughly 10,000 units per month at
an average price of $3,000 per unit. At the time, Apple claimed about a 9% market share
of the desktop market (down from greater than 15% in the 1980s).
By the end of 1995, Apple had witnessed a dramatic shift in the competitive environment.
In the preceding 18 months, Intel had cut the prices of its top-performing Pentium chip by
some 40%. Consequently, Apple’s two largest competitors, Compaq and IBM, reduced
average PC prices by 15%. Mail-order retailer Dell continued to gain market share via
aggressive pricing. At the same time, Microsoft introduced Windows 95, finally offering
the PC world the look and feel of the Mac interface. Many software developers began
producing applications only for the Windows operating system or delaying development
of Macintosh applications until months after Windows versions had been shipped.
Overall, fewer users were switching from PCs to Macs.
Apple’s top managers grappled with the appropriate pricing response to these competitive
events. Driven by the speedy new PowerPC chip, the Power Mac offered capabilities and
a user-interface that compared favorably to those of PCs. Analysts expected that Apple
could stay competitive by matching its rivals’ price cuts. However, John Sculley, Apple’s
CEO, was adamant about retaining a 50% gross profit margin and maintaining premium
prices. He was confident that Apple would remain strong in key market segments – the
home PC market, the education market, and desktop publishing.
Questions.
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1. What effect (if any) did the events of 1995 have on the demand curve for Power Macs?
Should Apple preserve its profit margins or instead cut prices?
2. a) In 1994, the marginal cost of producing the Power Mac was about $1,500 per unit,
and a rough estimate of the monthly demand curve was: P = 4,500 – .15Q. At the time,
what was Apple’s optimal output and pricing policy?
b) By the end of 1995, some analysts estimated that the Power Mac’s user value (relative
to rival PCs) had fallen by as much as $600 per unit. What does this mean for Apple’s
new demand curve at end-of-year 1995? How much would sales fall if Apple held to its
1994 price? Assuming a marginal cost reduction to $1,350 per unit, what output and price
policy should Apple now adopt?
1 This account is based on J. Carlton, “Apple’s Choice: Preserve Profits or Cut Prices,”
The Wall Street Journal, February 22, 1996, p. B1.
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ADDITIONAL MATERIALS
I. Readings
A. Lowry, “Is Uber’s Surge-Pricing an Example of High-Tech Gouging?”
The New York Times Magazine, January 12, 2014, pp. 18-20.
J. Jargon and E. Glazer, “Crisis Quickens at Quiznos,” The Wall Street
Journal, December 7, 2013, p. B1.
J. Jargon, “Battle over Menu Prices Heats Up,” The Wall Street Journal,
August 23, 2013, p. B6.
R. H. Frank, “How can they Charge that? (and other questions),” The New
York Times, May 12, 2013, p. BU8.
A. Gasparro, “Amid Falling Profit, McDonald’s to Revisit ‘Dollar Menu’,”
The Wall Street Journal, October 20, 2012.
M. Kaminski, “An Airline that Makes Money, Really,” The Wall Street
Journal , February 4, 2012, p. A13.
M. Cieply, “For Movie Stars, the Big Money is now Deferred,” The New
York Times, March 4, 2010, p. A1.
N. S. Riley, “Other People’s Money,” The Wall Street Journal, October 3,
2008, p. W11.
R. Chittum, “Price Points,” The Wall Street Journal, October 30, 2006, p.
R7. (How providers of consumer services compare extra revenues and extra
costs.)
T. H. Davenport, “Competing on Analytics,” Harvard Business Review,
January 2006.
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C. Oggier and E, Fragniere, and J. Stuby, “Nestle Improves its Financial
Reporting with Management Science,” Interfaces, July-August, 2005, pp.
271-280.
II. Case
Colgate-Palmolive Co.: The Precision Toothbrush (9-593-064), Harvard
Business School, 1993. Teaching Note (5-595-025). (Explores profit
analyses of alternative launch strategies.)
III. Quips and Quotes
Small mistakes are the stepping stones to large failures.
There was an old saying about our small town. Our town’s population never
changed. Every time a baby was born a man left town. (Does this say
something about the balance of marginal changes at an optimum?)
The head of a small commuter plane service reported that as costs rose, the
company’s breakeven point rose from 6 to 8 to 11 passengers. “I finally
figured we were in trouble since our planes only have 9 seats.”
If you laid all of the economists in the world end to end, they still wouldn’t
reach a conclusion. (George Bernard Shaw)
An economist is a person who is very good with numbers but who lacks the
personality to be an accountant.
The age of chivalry is gone; that of sophisters, economists, and calculators
has succeeded. (Edmund Burke)
Please find me a one-armed economist so we will not always hear, “On the
other hand . . .” (Herbert Hoover)
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