Academic journal article Journal of Research in Educational Sciences

Solving Optimal Timing Problems Elegantly

Academic journal article Journal of Research in Educational Sciences

Solving Optimal Timing Problems Elegantly

Article excerpt

Abstract:

Few textbooks in mathematical economics cover optimal timing problems. Those which cover them do it scantly or in a rather clumsy way, making it hard for students to understand and apply the concept of optimal time in new contexts. Discussing the plentiful illustrations of optimal timing problems, we present an elegant and simple method of solving them. Whether the present value function is exponential or logarithmic, a convenient way to solve it is to convert the base to the exponential number e, thus making it easy to differentiate the new objective function with respect to time t. This convenient method of base conversion allows to find a second-order derivative and to use the second-order condition as a proof of optimum.

Keywords: optimization of functions of one variable, continuous time, optimal timing, discounted present value, future value.

(ProQuest: ... denotes formulae omitted.)

1. Introduction

Optimal timing problems represent an interesting set of illustrations related to economic dynamics and the role of the time factor in economic decision making. These are economic applications of univariate calculus where the argument is time and, given a specific objective function, usually the net present value of an asset or an economic resource, the optimal time or period of appropriating or harvesting the respective resource must be found which maximizes that value. These are problems which answer the question when is the best time, i.e., when is it best to pick up the resource, to harvest the crops or to sell the asset the value of which appreciates with time. Resources could generally be classified as appreciating or depreciating in value terms. Optimal timing problems study resources whose value appreciates with time in terms of rate of growth. When an economic resource loses value with the passage of time, that is, its discounted present value falls, optimal timing problems demonstrate its depreciation in the context of its rate of decay.

Various illustrations could be given of resources whose value grows as time goes by. Examples of this type answer questions such as when is the best time to pick olives, oranges, or peaches in the orchard, tomatoes in the garden, flowers in a flower plant or a greenhouse, grapes in the vineyard so that to produce wine. Except crops such examples ask when it is best to cut trees so that to maximize the value of lumber or to receive highest yields from selling it. Also from the realm of environmental and natural resource economics we may seek to find when to harvest fish or other from a common-pool or a common access resource. Optimal timing problems have relevance not only to mathematical but to environmental economics as well. Examples from mining and extraction depict resources whose value depreciates with time, for instance, extraction of oil from an oil well, mining from mines, etc. With extraction and mining the value of the resource will be declining with time as the resource gets depleted and it becomes harder to extract it.

Examples of appreciating assets beyond the scope of agricultural and environmental economics include artifacts, collection items and jewelry. Such examples ask when the best time is to sell a ruby, a Picasso picture, or a golden Rolex. In view of their diverse illustrations it is puzzling that most economic textbooks ignore optimal timing problems or present them mostly or only in the context of financial economics. The usual representation in economic literature is that of the principal and the interest where the discounted value of a financial asset (say a deposit) is sought to be maximized. The net present value can be traced using either a given simple or a compounded interest. This limited framework within which optimal timing is presented makes it uninteresting for undergraduate students who cannot always see the diverse and plentiful illustrations that optimal timing problems pose. The paper studies the essence, importance, validity and variety of optimal timing problems as they are covered in mathematical economics. …

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