The economics of purifying and blending.

• Author: Hennessy, David A.

1. Introduction

   Purification and blending are processes widely used in a broad variety of industries. Oil is refined and mixed. Bulk chemicals, such as alcohols, dyes, salts, fertilizers, and cements, have impurities removed, and are homogenized. Biochemicals, beverages, fruits, vegetables, and grains are also subjected to similar transformations, as are bulk and precious metals.

   While programming methods have been used for many years to provide practical answers to these processing decision problems [4], little analytical work has been done to identify which processes are appropriate, and how policy can affect processing decisions. Though a literature on screening for quality exists [12; 13], these models deal with information gathering and asymmetric information issues. So too do the studies of market failures due to quality uncertainty [1; 9; 10]. Kenney and Klein [8] consider how asymmetric information motivates the packaging of qualitatively heterogeneous goods to avoid search costs, while Barzel [2] identifies practices that have arisen due to measurement problems. Although these papers do deal with concepts similar to purification (i.e., quality certification) and blending (i.e., average quality), the actions of decision-makers and policy-makers are motivated by information set differences. In the model presented below, all agents have complete information.

   An economic analysis of purification and blending is important because regulations mandating or banning these transformations are often enacted. Such legislation is particularly prevalent for drugs, agricultural produce, petroleum products, gases, and precious metals. These regulations may be established for a variety of reasons such as safety (gases, food, flammable liquids), sectoral income support (food), and fraud prevention (precious metals). The purpose of this paper is to establish a framework within which to analyze normative decision-making and prescribe regulatory policy.

   Models of purification and blending are first established. Optimal decisions are then identified, and their effects on the value of products along the quality spectrum are described. The effects of processing costs and policy decisions on equilibrium are then demonstrated.

2. Purification

   Consider a commodity of mixed quality which can be separated completely into high quality and low quality constituents. Associated with each unit of the mixed quality commodity is a quality index, x, which ranges over the interval [0, 1]. Here, 0 and 1 are the lowest and highest qualities, respectively. The commodity is mixed in the sense that all units are mixtures of high quality (quality = 1) and low quality (quality = 0) constituents. The fraction of a unit that is high quality gives the quality measure. Thus, a unit of quality [x.sub.c] is comprised of 100[x.sub.c]% quality 1 material and 100(1 - [x.sub.c])% quality 0 material, and can be separated into [x.sub.c] units of quality 1 material and 1 - [x.sub.c] units of quality 0 material. The value of a unit of quality x when retailed in that form is given by P(x), a continuous, increasing function. When this function is evaluated at a specific value, say [x.sub.c], it is written as P([x.sub.c]). This price-quality relationship shall be called the raw price-quality schedule from now on because it is the price that quality x would command if all processing were banned. While this function could plausibly have many shapes, without loss of generality it is assumed here to be convex at low values of x, and concave at high values of x in the interval [0, 1]. For wheat grade standards that possess approximately this shape, see Hyberg et al. [7]. Other shapes will be discussed later.

   The function is illustrated in Figure 1. The intersection on the vertical axis is denoted by the cartesian point (0, P(0)). It is clear from the figure that commodity units with quality index values in the convex region may benefit from some purification, i.e., separation into quality 0 material and material of quality at least as great as the point of inflexion if purification is costless. The point of inflexion, denoted by [x.sub.I], is the quality level at which the raw price-quality schedule changes from being convex to concave. That the raw material would benefit from some purification in the convex region arises naturally from Jensen's inequality which states that for a P(x) function that is convex over [a, b], and for an arbitrary measure with mass density function f(x),

   [integral of] P(x)f(x)dx between limits b and a [greater than or equal to] P ([integral of] xf(x)dx between limits b and a). (1)

   Here the mass density function of x denotes how quality is distributed after the purification. The distribution is constrained by the fact that mean quality must not change. Equation (1) says that in the convex region the market provides an incentive to separate the good from the bad because the total value of the parts (i.e., the value of purified materials) exceeds the value of the unpurified mixture.

   To provide a more analytical foundation to this conclusion, consider a unit measure of quality [x.sub.c] in the convex portion of the schedule in Figure 1. Under purification it will be separated into [x.sub.c]/[x.sub.*] units of quality [x.sub.*], and 1 - ([x.sub.c]/[x.sub.*]) units of quality 0. Here, [x.sub.*] is a yet to be determined quality that is greater than [x.sub.c]. The possibility of [x.sub.*] being in the concave region cannot be precluded. Let there be a fixed cost, K, associated with the purification process. Also, let the variable costs of purification be proportional to the increase in quality of the purified component, [x.sub.*] - [x.sub.c]. Therefore, the total cost of purification is s([x.sub.*] - [x.sub.c]) + K where s is the unit marginal cost of purification.(1) The problem facing a price-taking processor is to find the quality [x.sub.*] to which [x.sub.c] should be purified by maximizing the per unit net revenue after purification has occurred,

   ...