Chapter 4 The Theory of Optimization
In this chapter we will give
you the key to the kingdom of
economic decision making:
marginal analysis.
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
Virtually all of microeconomics involves solutions
to optimization problems. The most interesting
and challenging problems facing a manager
involve trying either to maximize or to minimize
particular objective functions. Regardless of
whether the optimization involves maximization or
minimization, or constrained or unconstrained
choice variables, all optimization problems are
solved by using marginal analysis.
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 We
begin the analysis of optimization
theory by explaining some
terminology.then derive two rules for
making optimal decisions.
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4.1CONCEPTS AND TERMINOLOGY

objective function
The function the decision maker seeks to
maximize or minimize.
 maximization problem
An optimization problem that involves maximizing
the objective function.
 minimization problem
An optimization problem that involves minimizing
the objective function.
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Activities or Choice Variables

activities or choice variables
Determine the value of the objective function.
 discrete choice variable
A choice variable that can take only specific
integer values.
 continuous choice variable
A choice variable that can take on any value
between two end points.
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unconstrained and constrained optimization

unconstrained optimization
An optimization problem in which the decision
maker can choose the level of activity from an
unrestricted set of values.
 constrained optimization
An optimization problem in which the decision
maker chooses values for the choice variables
from a restricted set of values.
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
constrained maximization
A maximization problem where the activities must
be chosen to satisfy a side constraint that the
total cost of the activities be held to a specific
amount.
 constrained minimization
A minimization problem where the activities must
be chosen to satisfy a side constraint that the
total benefit of the activities be held to a specific
amount.
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marginal analysis

An analytical tool for solving optimization
problems that involves changing the value(s)
of the choice variable(s) by a small amount
to see if the objective function can be
further increased (for maximization
problems) or further decreased (for
minimization problems).
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4.2 unconstrained maximization
 The
results of this chapter fall neatly
into two categories: the solution to
unconstrained and the solution to
constrained optimization problems.
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
When the values of the choice variables are not
restricted by constraints such as limited income,
limited expenditures, or limited time, the
optimization problem is said to be unconstrained.
 One of the most important unconstrained
optimization problems facing managers is
selecting the set of variables that will maximize
the profit of the firm.
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
This problem and all other unconstrained
maximization problems can be solved by
following this simple rule: To maximize an
objective function, the value of which
depends on certain activities or choice
variables, each activity is carried out until
the marginal benefit from an increase in the
activity equals the marginal cost of the
increased activity:
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
MBA=MCA,MBB=MCB,…..,MBZ=MCZ

optimal level of the activity


The level of activity that maximizes net benefit.
marginal benefit (MB)
The addition to total benefit attributable to increasing the
activity by a small amount.
marginal cost (MC)
The addition to total cost attributable to increasing the
activity by a small amount.
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
When the choice variables are not
continuous but discrete, it may not be
possible to precisely equate benefit and
cost at the margin. For discrete choice
variables, the decision maker simply carries
out the activity up to the point where any
further increases in the activity result in
marginal cost exceeding marginal benefit.
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Sunk costs and fixed costs are irrelevant
sunk costs
Costs that have previously been paid and
cannot be recovered.
 fixed costs
Costs that are constant and must be paid
no matter what level of the activity is
chosen.

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

中国航空工业第一集团公司在2000年8月决定今后民用飞机
不再发展干线飞机，而转向发展支线飞机。这一决策立时
引起广泛争议和反弹。
许多人反对干线飞机项目下马的一个重要理由就是，该项
目已经投入数十亿元巨资，上万人倾力奉献，耗时六载，
在终尝胜果之际下马造成的损失实在太大了。这种痛苦的
心情可以理解，但丝毫不构成该项目应该上马的理由，因
为不管该项目已经投入了多少人力、物力、财力，对于上
下马的决策而言，其实都是无法挽回的沉没成本。
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
事实上，干线项目下马完全是“前景堪忧”使
然。从销路看，原打算生产150架飞机，到
1992年首次签约时定为40架，后又于1994
年降至20架，并约定由中方认购。但民航只
同意购买5架，其余15架没有着落。可想而
知，在没有市场的情况下，继续进行该项目
会有怎样的未来收益？
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
沉没成本与人生态度
经济学中有许多概念不仅有利于经营企业，而
且对于认识人生也是有益的。沉没成本这个概念是
其中之一。
当一项业已发生的成本，无论如何努力也无法
收回的时候，这种成本就构成了沉没成本。面对这
种无法收回的沉没成本，明智的投资者会视其为没
有发生。举个例子来说，你花了10块钱买了一张今
晚的电影票，准备晚上去电影院看电影，不想临出
门时天空突然下起了大雨。这时你该怎么办？
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4.3 constrained optimization

In many instances, managers face limitations on
the range of values that the choice variables can
take. For example, budgets may limit the amount
of labor and capital managers may purchase.
Time constraints may limit the number of hours
managers can allocate to certain activities. Such
constraints are common and require modifying
the solution to optimization problems.
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
To maximize or minimize an objective
function subject to a constraint, the ratios of
the marginal benefit to price must be equal
for all activities,

MBA/PA=MBB/PB=……MBZ/PZ
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
and the values of the choice variables must
meet the constraint. One of the most
important constrained optimization
problems facing a manager is the task of
producing a given output at the least
possible total cost.
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Case study
Is cost-benefit analysis really useful?
(pp134)
? Find the answers to question
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