The economics of purifying and blending.

I. 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.

II. 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,

[Mathematical Expression Omitted].

The first order condition is

[Delta]P/[Delta]x = [P(x) - P(0)]/x + sx/[x.sub.c], (3)

the solution of which is x = [x.sub.*].(2) If purification occurs, and if there is perfect competition among processors, then profit from purifying equals zero and the price that purifiers are willing to pay for raw material with quality [x.sub.c] is equal to the valuation of the objective function given in equation (2) at x = [x.sub.*].

Note that [P(x) - P(0)]/x measures the average value of quality (AVQ) at quality level x. P(0) has been removed when evaluating AVQ because this component of price does not pertain to the quality index x. Equation (3) implies that a positive marginal cost of purifying guarantees that [x.sub.*] is less than the x which equates AVQ with marginal value of quality (MVQ), where MVQ is the tangent to P(x). The solution to AVQ = MVQ is an important point in this analysis, and is denoted by [x.sub.m]. [x.sub.m] must be in the concave portion of the curve because the AVQ for the increasing function under consideration cannot equal MVQ at any point on the convex portion of the function. It is possible that no interior solution exists, i.e., that [x.sub.m] = 1. If this corner solution holds, then it is also possible that [x.sub.*] = 1 for low values of s. It is not, however, assured that [x.sub.*] is in the concave region. The solutions to equation (3) and to AVQ = MVQ are depicted in Figure 1. An important insight from equation (3) is that [x.sub.*] depends on [x.sub.c]. This dependency occurs through the marginal cost of purifying. Note that if s = 0, then [x.sub.*] = [x.sub.m], and is independent of [x.sub.c]. Comparative statics of equation (3) give

[Delta][x.sub.*]/[Delta][x.sub.c] = -[[Delta]P([x.sub.*]/[Delta]x - (P([x.sub.*]) - P(0))/[x.sub.*]]/[[x.sub.c][[Delta].sup.2]P([x.sub.*])/[Delta][x.sup.2] - 2s], (4)

[Delta][x.sub.*]/[Delta]s = [x.sub.*]/[[x.sub.c][[Delta].sup.2]P([x.sub.*])/[Delta][x.sup.2] - 2s]. (5)

The numerator inside the brackets in (4) is positive, while the denominator is assuredly negative if [x.sub.*] is in the concave region. Thus, if [x.sub.*] is in the concave region then expression (4) is positive and expression (5) is negative. This means that, if [x.sub.*] is in the concave region, then the quality of the processed material increases with the quality of the raw material, and decreases with the marginal cost of purification.

The above analysis has not taken account of fixed costs. If, at the optimum x,

[1 - ([x.sub.c]/x)]P(0) + ([x.sub.c]/x)P(x) - s(x - [x.sub.c]) - P([x.sub.c]) - K [less than] 0, (6)

then quality [x.sub.c] will not be purified. This may occur at [x.sub.c] values close to [x.sub.m], where the value added from purifying may be less than K. The inequality may also occur at [x.sub.c] values close to 0, where the total cost of purifying, s([x.sub.*] - [x.sub.c]) + K, is prohibitive. In the absence of blending, unpurified units have value given by the raw price-quality schedule.(3)

To summarize, there is a quality interval in the neighborhood of [x.sub.m] where purification does not occur because the value added from purification does not cover fixed costs, and there is also a quality interval in the neighborhood of 0 where purification may not occur because value added does not cover total costs. Purification need not be complete ([x.sub.*] [less than or equal to] 1 with the strict inequality possible), and will not be complete if ever AVQ and MVQ are equal on [0, 1). The degree of purification depends on the raw quality of the load, and the marginal cost of purification. Further, if purification occurs and [x.sub.*] is in the concave region, then the degree of purification increases with raw quality and decreases with the marginal cost of purification.

However, purification cannot be considered in isolation from the blending process. This will be demonstrated in section IV. But first, the pure blending process is considered.

III. Blending

First, consider only a concave raw price function as depicted in Figure 2, i.e., for the moment all raw qualities are assumed to fall in the interval where the function is concave. Then, if blending is costless, Jensen's inequality implies that maximum value added occurs when all units are blended. If raw quality is distributed with mass density f(x) over [a, b], then the price received for all blended units is P([integral of] xf(x)dx between limits b and a). For the units with initial quality equal to mean quality, [Mathematical Expression Omitted], price does not change as a consequence of blending because blending does not alter the unit's composition, i.e., even in the absence of blending costs there is zero value added. However, with no blending costs, price at mean quality provides a marginal value per unit quality. Take a Taylor's series expansion of P(x) around [Mathematical Expression Omitted] to obtain

[Mathematical Expression Omitted],

the tangent line in Figure 2. Here, the V(x, 0) line is the exact value of raw material when blending is both possible and permitted, and when blending costs are zero. The second argument in the function V([center dot], [center dot]) represents the per unit blending cost. The V(x, 0) function gives the exact value of raw material for all values of x because the Taylor's series expansion is not an approximation away from the point of expansion.

The linearity of V(x, 0) arises from the innate linear homogeneity [6] of the blending process. To illustrate this linear homogeneity property, a unit of quality [x.sub.d] can be decomposed into [x.sub.d] units of quality 1 and 1 - [x.sub.d] units of quality 0, while a unit of quality [x.sub.e] can be decomposed into [x.sub.e] units of quality 1 and 1 - [x.sub.e] units of quality 0. Let [Alpha] be a number in [0, 1]. Linear homogeneity is satisfied because one unit of quality [Alpha][x.sub.d] + (1 - [Alpha])[x.sub.e] can be decomposed into [Alpha][x.sub.d] + (1 - [Alpha])[x.sub.e] units of quality 1 and 1 - [Alpha][x.sub.d] - (1 - [Alpha])[x.sub.e] units of quality 0. But [Alpha] units of quality [x.sub.d] can be decomposed into [Alpha][x.sub.d] units of quality 1 and [Alpha](1 - [x.sub.d]) units of quality 0, while 1 - [Alpha] units of quality [x.sub.e] can be decomposed into (1 - [Alpha])[x.sub.e] units of quality 1 and (1 - [Alpha])(1 - [x.sub.e]) units of quality 0. These latter decompositions sum to [Alpha][x.sub.d] + (1 - [Alpha])[x.sub.e] units of quality 1 and 1 - [Alpha][x.sub.d] - (1 - [Alpha])[x.sub.e] units of quality 0, the same as the decomposition of a unit of quality [Alpha][x.sub.d] + (1 - [Alpha])[x.sub.e]. It is because the decomposition is independent of convex combinations that the value line is linear. Note that while units close to mean quality benefit only slightly from the blending process, the more dispersed units may benefit significantly.

Now consider the more realistic situation where cost is incurred in blending. Let the blending cost be B per unit. The situation is illustrated by the line beneath and parallel to the tangent line in Figure 2. The parallel structure arises because per unit blending costs are invariant to raw quality. The vertical distance between the two lines is B. Now, not all units in the interval [a, b] can cover the cost of blending, and units close to [Mathematical Expression Omitted] will not be blended. The raw qualities of these units are too close to the ultimate blended quality to benefit significantly from exploiting the concavity of the price-quality schedule. Upon excluding these central units, mean quality, and so mean blend price, will change, i.e., the V(x, B) function after taking B [greater than] 0 into account need not be parallel to the V(x, 0) function. Let [a.sub.1] and [b.sub.1] be, respectively, the lower and upper qualities where V(x, B) intersects P(x). These points may be solved for by noting that they are marginal units where the processor is indifferent between blending and not blending. The quality of the blended units, [Mathematical Expression Omitted], is given by

[Mathematical Expression Omitted]

Here, the non-blended units have been omitted and the mass density function has been scaled up accordingly. Mean price and MVQ when blending occurs are, therefore, given by [Mathematical Expression Omitted] and [Mathematical Expression Omitted], respectively. Now it is possible to identify prices where blending does not make a difference. In particular, the constant MVQ result in (7) can be used to develop the break-even equations

[Mathematical Expression Omitted],

[Mathematical Expression Omitted].

These equations together with equation (8) solve for [a.sub.1], [b.sub.1], [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted].

The resulting value of raw material function if blending is allowed is depicted by the heavy line in Figure 3, and is denoted by Z(x, B). It can be seen that a ban on blending would not affect average quality producers, but would affect the more dispersed quality producers. It is worth considering what effect the distribution of quality has on the benefits from blending. If blending is banned, then, because of the increasing and concave nature of P(x), any second degree stochastically dominating change (SSD) in the distribution of x increases the mean value of the raw material [5]. To show this result, let the cumulative mass density function G(x) dominate F(x) in the SSD sense. If both have supports on [a, b], then it is only necessary to show that

[integral of] P(x)d[G(x) - F(x)] between limits b and a [greater than] 0. (11)

Integrate by parts to obtain

[Mathematical Expression Omitted].

On the right hand side of the last equality sign, the first integrand and the inner integral of the second integration are positive due to the SSD ranking. Thus the expression is positive if, as is assumed, P(x) is increasing and concave.

If blending is permitted and B = 0, then the linearity of V(x, B) means that no SSD can decrease the mean value of raw material. However, because mean quality increases under any SSD that is not a mean preserving contraction, and because of the concavity of P(x), a mean increasing SSD decreases MVQ. If blending is costly, so that the blended price-quality schedule is like that in Figure 3, then any first degree stochastically dominating (FSD) change in x increases mean value of raw material [5]. This is because an FSD increases the mean value of any increasing function, and Z(x, B) is increasing in x. However, as Z(x, B) is not concave in x, nothing can be said about SSD changes in the quality distribution.

Another issue that merits some consideration is that of quality testing. Suppose that processing is prohibited, and the producer sells the raw material directly to a retailer. The retailer tests for quality, and pays accordingly. If the quality test is correct on average, but is subject to an additive random error, [Xi], where the expected value of [Xi] is zero for all values of x, then the producer is underpaid when P(x) is concave [11]. However, if blending is permitted and costless, then the producer receives a fair price on average because of the linearity of V(x, 0).

So far the two processes have been considered separately. In the next section they are brought together.

IV. Integrating the Processes

Consider again the raw price-quality schedule in Figure 1 which is increasing everywhere, convex close to 0, and concave close to 1. In the first part of this section a general discussion about integrating the processes when there are processing costs is provided. The complexity of finding a solution is demonstrated. In the second part, processing costs are assumed away, and a precise analysis of integrating the processes is performed.

Suppose for the moment that if x [less than] [x.sub.m], then either purification or no processing occurs, while if x [greater than] [x.sub.m] then either blending or no processing occurs. The objective function in equation (2), condition (3), and inequality (6) together give a value after purification function which will be denoted by W(x, K, s). As explained previously, at low values of x and at values of x close to [x.sub.m], the equality W(x, K, s) = P(x) may hold. Let [Mathematical Expression Omitted] be the quality level of blended units. Now consider whether or not units with raw quality in [0, [x.sub.m]] will be blended after the purification decision has been made. Blending occurs for an x, purified or not, in the interval [0, [x.sub.m]] if

[Mathematical Expression Omitted].

That is, blending occurs if value net of B increases due to blending. As the units already blended into the units of quality [Mathematical Expression Omitted] are in the interval [[x.sub.m], 1], the introduction of new material into the blend decreases [Mathematical Expression Omitted] and increases the slope of Z(x, B) in the regions where it is linear. It also decreases [a.sub.1] and [b.sub.1]. Because, from equations (8) through (10), [Mathematical Expression Omitted], [a.sub.1], and [b.sub.1] change as new material is blended, sequencing matters. Quality levels where inequality (13) is most severe should be introduced first into the blend, for otherwise it may prove profitable to purify or to refrain from blending some of the blend later. In other words, some of the material in the convex region that seems to warrant blending may not warrant blending after other material has been added to the blend. Similarly, some of the material in the concave region initially slated for blending (for leaving alone) may not (may) warrant blending after material initially in the [0, [x.sub.m]] interval has been added to the blend.

Apart from the above mentioned sequencing problem, there is another problematic sequencing issue. The discussion above assumes that the higher quality raw units are initially designated candidates for blending, and the lower quality raw units are initially designated candidates for purification. However, as it may happen that purified units might be subsequently blended, costs might be reduced by forgoing the purification of lower quality units and blending them directly. The ultimate solution of these sequencing problems might best be found by trial and error mathematical programming.

To simplify the analysis so as to better understand the economics of the processing operations, let processing costs be zero, i.e., let B = K = s = 0. Let Y(x) denote the value after processing of a unit of raw quality x. As before, let [Mathematical Expression Omitted] be the mean quality of all units. There are now two cases, and both are illustrated in Figure 4. The first is where [Mathematical Expression Omitted]. Denote it by [Mathematical Expression Omitted] in Figure 4. In this case, Y(x) is given by the straight line joining (0, P(0)) and ([x.sub.m], P([x.sub.m])). This price line can be achieved by blending all units and then purifying until the purified material is of quality [x.sub.m].

As processing is costless, the sequencing of processes is irrelevant. The processor will retail a fraction [Mathematical Expression Omitted] of total units at price P([x.sub.m]), and will retail the residual fraction, [Mathematical Expression Omitted], of total units at price P(0). This gives an average unit value (AUV) of [Mathematical Expression Omitted], while MVQ and AVQ equal [P([x.sub.m]) - P(0)]/[x.sub.m]. AUV is invariant to a mean preserving contraction in quality, but increases under any other SSD. MVQ and AVQ are invariant to any SSD that maintains [Mathematical Expression Omitted]. Producers of raw material will not command prices better than Y(x) because if a processor pays more for a high (low) quality raw unit, he cannot purchase sufficient low (high) quality raw material to blend with it and still break even.

The second case is where [Mathematical Expression Omitted]. Denote it by [Mathematical Expression Omitted] in Figure 4. In this case, there is so much high quality raw material that there is a deficiency in low quality raw material to blend with it. Now, as in section III above, Y(x) is given by the tangent to P(x) at [Mathematical Expression Omitted]. By construction, this intersects the vertical axis at a value greater than P(0). Thus, the deficiency of love quality raw material gives rise to a higher price for quality zero material than when [Mathematical Expression Omitted]. As in the case where [Mathematical Expression Omitted], AUV is invariant to any mean preserving contraction in quality. Also, like the previous case, AUV increases under any other SSD transformation of quality. However, unlike the previous case, AUV is concave in [Mathematical Expression Omitted], rather than linear. Also, unlike the [Mathematical Expression Omitted] case, MVQ decreases under an SSD in quality because an SSD increases [Mathematical Expression Omitted] which results in a decrease in [Mathematical Expression Omitted] due to concavity. AVQ also decreases under an SSD because [x.sub.m] gives maximum AVQ, and because of the concavity of P(x) in [[x.sub.m], 1].

V. Discussion

This paper looks at two production operations that are pervasive in industry. Curvature of the raw material price-quality schedule is shown to motivate the operations. The case where price is convex in quality at low quality levels, but concave in quality at higher qualities, is examined. When processing is costly, some intervals in the quality spectrum may be blended and some may be purified, while several intervals throughout the spectrum may remain unprocessed. When processing is costless, the quality where AVQ equals MVQ is demonstrated to have special significance. It is interesting to note that this point together with the optimal processing decisions have analogues in financial portfolio theory when there is a risk-free asset with rate of return equal to P(0) [3]. Perhaps this analogy warrants further research.

Other price-quality relationships are also possible. For example, P(x) might be concave at low quality levels but convex at higher quality levels. Here, the blending and purification processes would also be interlinked, but in a different way. A thorough analysis would involve a case by case study as in section IV. Another possible schedule is an increasing, discontinuous, step function price-quality relationship. Such functions are commonly encountered when grading categories and minimum quality standards exist. They are neither concave nor convex. Points of discontinuity will be of particular importance in the analysis, and a programming approach may be most appropriate in identifying optimal processing decisions.

The paper does not demonstrate any instances where it is optimal for anyone to seek a ban on processing operations. However, this is because no externalities have been modeled for. Externalities might occur if quality tests are expensive or inaccurate, and if trading partners are not engaged in a long-term trading relationship. Due to advances and cost reductions in chemical and physical testing, and due to the relatively concentrated nature of many bulk materials industries, such information externalities may not often occur in these industries. In agriculture, where there are generally a large number of sources for raw materials, externalities may be more common.

1. Other cost specifications are also possible. For example, technology will determine whether to screen out the good from the bad or the bad from the good. Also, technology might be such that the lower quality units in the separation have index value greater than zero. This would present a slightly more difficult problem than the one analyzed here.

2. Equation (2) gives

([x.sub.c]/[x.sup.2]) [P(0) - P(x)] + ([x.sub.c]/x)[Delta]P/[Delta]x = s,

which can be rearranged to give equation (3). The second order condition is

(-2[x.sub.c]/[x.sup.3]) [P(0) - P(x)] - 2([x.sub.c]/[x.sup.2])[Delta]P/[Delta]x + ([x.sub.c]/x)[[Delta].sup.2]P/[Delta][x.sup.2] = -2(s/x) + ([x.sub.c]/x)[[Delta].sup.2]P/[Delta][x.sup.2].

This is negative if P(x) is concave at [x.sup.*].

3. The case where blending is permitted is examined in section IV.

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