Some Notes on "Dynamic" Linear Programming

The most common application of linear programming in agricultural situations has been to the problem of resource allocation between competing farm activities. Given relevant input-output information for a specific farm, together with real or assumed price and cost patterns, the technique of linear programming enables calculation of the combination of enterprises which maximizes net profit, within the limitations imposed by the availability of farm resources. It is necessary in some linear programming analyses to make explicit allowance for the peculiar influence of time on the structure of the system under study. Of the many ways in which this may be achieved, this article considers four, which have been, or are likely to be, of relevance in an agricultural context: (i) Parametric programming, which allows consideration of resource or price variation between time periods; (ii) extension of the time-span of an activity to cover a series of sequential processes, for example the treatment of rotational sequences as single activities; (iii) the referencing of some resources and/or activities to specific time periods; a common example is the fragmentation of labour supply into months; and (iv) the so-called "multi-stage" or "dynamic" linear programming where a single matrix is used to describe, in an orderly fashion, a system's structure over a time-span of several periods. It is the latter with which we are primarily concerned here. In its simplest form a dynamic linear programming problem may be set up as a large matrix composed of a series of smaller matrices lying down the diagonal. In its more advanced form allowance can be made for interactions between resources and activities in different periods. In general, dynamic linear programming problems are characterized by large "sparse" matrices (i.e., matrices in which many coefficients are zero) and usually a "block diagonal" or "block triangular" pattern is evident. The size of such matrices is frequently forbidding; however, computational algorithms are available which allow overall solutions to be obtained by solving a series of smaller problems. With the aid of a little ingenuity a great variety of time-dependent restrictions, resources, activities and opportunities can be accounted for in a dynamic linear programming analysis. From an agricultural economist's viewpoint it would not seem extravagant to claim that dynamic linear programming can be used to provide a more adequate analytical description of whole-farm situations over time than most other tools at present available in his kit

STATIONARY-STATE SOLUTIONS IN MULTI-PERIOD LINEAR PROGRAMMING PROBLEMS

Multi-period linear programming models are classified initially. The main interest of this paper is in models of cyclical activity. The conditions under which such models lead to stable repetitive optimal solutions are defined. A simple technique is presented by which this stable solution (termed here the "stable core") can be derived without recourse to solving a full extended matrix. Firm-level applications of this model in agriculture are discussed

GAME THEORY AND A TIME-OF-MARKET DECISION

The contribution of the theory of games to decision making has been twofold. Firstly, it has described and clarified many of those decision complexes which are reducible to a direct conflict of interests ("game") between two or more irreconcilably opposing parties who have some control over the outcome of the conflict. Secondly, it has supplied means of obtaining solutions to some of the problems so posed. The best examples of non-trivial real-world game situations are to be found in warfare and, in the field of business, in duopolies and oligopolies. But the number of applications has been limited to a large extent by some assumptions required by the theory, mainly those which define the strictness of the competition between the players and the degree of knowledge each has about the others' strategies. In agriculture, too, it is rare that these assumptions can be rigorously met. However by adopting a rather liberal interpretation of these conditions, it is possible to conceive of some farm situations in terms of the theory; landlord vs. tenant or owner vs. sharefarmer competitions spring readily to mind, as do auction sales and other direct farmer-market contacts. One of the most fruitful areas for investigation in the agricultural field is the study of games against nature. This covers a range of situations, from direct conflict between the farmer and his physical environment, (for example, raising crops subject to irregular weather conditions) to a competition between the farmer and his economic environment, that is an amalgamation of all his "adversaries"--marketing authorities, other farmers, etc. In either case "nature" is considered as a fictitious player having no known objective and, as a starting point, no known strategy

Dynamic Programming, Activity Analysis and the Theory of the Firm

The role of dynamic programming as a means of examining the allocation and pricing problems in the theory of the firm is considered in this paper. The production relationships and equilibrium conditions as specified by neoclassical theory and linear programming are stated and dynamic programming formulations of each of these models are constructed and compared. It is demonstrated that dynamic programming adds nothing to established theory in these cases, providing simply an alternative means of computation which might be preferred for some empirical problems. It is concluded that some theoretical contribution may be possible by using dynamic programming to attack problems beyond the scope of conventional methods

Theoretical Aspects of a Dynamic Programming Model for Studying the Allocation of Land to Pasture Improvement

The study on which this article is based is hoped to be the first of a series investigating the use of dynamic programming in agricultural allocation decisions. The model presented in this article was constructed in simple form to study once-over sequences of allocation of land to some activity of long-run benefit. The activity chosen was pasture improvement, but as will be seen later the section of the model which describes pasture improvement could be altered to make the model specific to some other relevant activity if desired. The presentation of this article is as follows: First the nature of dynamic programming as an analytical tool is explained. Then the dynamic programming relationships in the model are constructed and a simple numerical example is given to illustrate the computational procedure. The second part of the model, which derives the pasture improvement return functions, is then shown. Some general computational problems are discussed, following which the properties of both parts of the model are investigated in some detail

Agriculture in the Economy: the evolution of economists' perceptions over three centuries

This paper traces the perceptions of the agricultural sector held by economists over the last three centuries, with particular emphasis on how the evolution of these ideas has influenced the state of present-day thinking about the economic role of agriculture in developed and developing economies. The paper begins with the seventeenth and eighteenth centuries, leading to a consideration of the place of agriculture in the work of the major figures of classical political economy from Adam Smith to Marx. The rise of neo-classical economics, and its influence on twentieth century thinking, is discussed. In the contemporary period, particular attention is paid to agriculture in development theory, with an assessment of conflicting theoretical ideas about the role of the agricultural sector during the process of economic transformation and growth. The paper concludes with a consideration of the current state of economic thought about the role of agriculture in the economy, and makes some observations on likely future directions