Investment and Disinvestment Principles with Nonconstant Prices and Varying Firm Size Applied to Beef-Breeding Herds: Comment

Optimal investment and disinvestment principles have received considerable attention in both general and agricultural economics for over two centuries. Notable are the works of Faustmann, Samuelson (1937), Hirschleifer, Burt, and Perrin which establish the basic replacement principle. Recently, Trapp presented a reformulation of this basic replacement principle by relaxing the assumptions of constant prices and firm size. The purpose of this comment is to discuss a possible misinterpretation of the replacement principle underlying Trapp's model. Specifically, the link between Trapp's article and previous literature is established with the hope of clarifying any misunderstandings of optimal investment principles. Trapp states that his equations (3) and (4), which are derived from the discrete time form of the net present value formula for a single asset, "constitute the appropriate addition and culling [investment and disinvestment] decision rules" for a firm with nonconstant size (Trapp, p. 694). This formulation is analogous to Perrin's equation (2), which maximizes the returns from an asset over a single rotation. In contrast, Chavas, Kliebenstein, and Crenshaw present the asset replacement decision as an optimal control problem with an infinite planning horizon. Transversality conditions for optimal asset rotation as given in Chavas, Kliebenstein, and Crenshaw's equation (12) correspond to Perrin's equation (4.2), which maximizes the returns from an asset over an infinite horizon. Reid and Bradford, in an investigation of investment and disinvestment in farm machinery, also derive optimal rotation criteria analogous to Perrin's formulation. Thus, it appears that an inconsistency exists in the Journal associated with the appropriate replacement rule when expansion and contraction of an operation is endogenously determined. Should replacement rules be derived from an infinite horizon specification or from a single asset specification? The merits of an investment decision must be based not only on earnings from the next production period but also on the rents which can be extracted from the firm's fixed factors. Failure to consider future assets in a replacement decision implies that changes in replacement costs have absolutely no effect on the optimal life of assets up to the point where it is no longer profitable to operate. Specifically, maximizing over a single asset rotation corresponds to maximizing the internal rate of return of an investment decision with a discount rate quivalent to the internal rate of return (Samuelson 1976). The quantitative effects of considering only a single asset specification are minimal depending on the length of the asset rotation and the discount rate employed. As the rotation length increases or a larger discount rate is employed, representation of the optimal replacement rule will converge to Perrin's equation (2) (Samuelson 1976). The turnpike theorem states that if the asset rotation is of sufficient length where the end of the horizon has o influence on period decisions, the system then converges towards the von Neumann path (Boussard). Too short a length may result in the system never converging toward the von Neumann path. Alternatively, if the salvage and acquisition values are equal at any point in time, then the von Neumann path will be reached at the beginning of the last period. However, in general, asset replacement is costly; and, thus, access to the von Neumann path is not instantaneous. The apparent inconsistent use of a single-asset specification in contrast to basic replacement principles associated with an infinite horizon may be resolved by investigating the theoretical results in Chavas and Klemme. Employing an optimal control methodology with a finite planning horizon, T, a distinction among rules derived from alternative specifications is evident. Using Perrin's terminology, Chavas and Klemme generalize the replacement criteria for a system with multiple constraints