Chapter 4 The Theory of Optimization

1. Chapter 4The Theory of Optimization In this chapter we will give you the key to the kingdom of economic decision making: marginal analysis.

2. 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.

3. We begin the analysis of optimization theory by explaining some terminology.then derive two rules for making optimal decisions.

4. 4.1CONCEPTS AND TERMINOLOGY • objective functionThe function the decision maker seeks to maximize or minimize. • maximization problemAn optimization problem that involves maximizing the objective function. • minimization problemAn optimization problem that involves minimizing the objective function.

5. Activities or Choice Variables • activities or choice variablesDetermine the value of the objective function. • discrete choice variableA choice variable that can take only specific integer values. • continuous choice variableA choice variable that can take on any value between two end points.

6. unconstrained and constrained optimization • unconstrained optimizationAn optimization problem in which the decision maker can choose the level of activity from an unrestricted set of values. • constrained optimizationAn optimization problem in which the decision maker chooses values for the choice variables from a restricted set of values.

7. constrained maximizationA 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 minimizationA 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.

8. 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).

9. 4.2 unconstrainedmaximization • The results of this chapter fall neatly into two categories: the solution to unconstrained and the solution to constrained optimization problems.

10. 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.

11. 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:

12. MBA=MCA,MBB=MCB,…..,MBZ=MCZ • optimal level of the activityThe 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.

14. 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.

15. Sunk costs and fixed costs are irrelevant • sunk costsCosts that have previously been paid and cannot be recovered. • fixed costsCosts that are constant and must be paid no matter what level of the activity is chosen.

16. 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。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。

17. 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？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？

18. 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？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？

19. 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.

20. 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

21. 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.

22. Case study Is cost-benefit analysis really useful? (pp134) ? Find the answers to question