Quality premiums and the firm's production decision.I. Introduction In real world production processes, the price-taking firm often produces a stream of products having varying quality. The firm may have limited control over the quality distribution, and may only observe quality after production decisions have been made. Often the marketplace can and does price output according to according to prep. 1. As stated or indicated by; on the authority of: according to historians. 2. In keeping with: according to instructions. 3. quality. Examples of the situations in question are beef, vegetable, fruit, and flower production, diamond mining, craft industries, and any production process where factory seconds exist. Another, though less apparent, example is where there is a minimum quality standard, a common practice in the production of branded goods. In this case, the price for production with quality below the standard is zero. While production under uncertainty [3; 5; 6; 10; 11] and hedonic pricing Hedonic Pricing A model identifying price factors according to the premise that price is determined both by internal characteristics of the good and external factors affecting it. [1; 8] have received considerable attention, the issue of production when faced with a price-quality schedule has not. In this paper, we adapt the stochastic dominance The term stochastic dominance is used in decision theory to refer to situations where one lottery (a probability distribution over outcomes) can be ranked as superior to another. It is based on preferences regarding outcomes (e.g., if each outcome is expressed as a number, e.g. approach [6, 430; 9, 227] to determine the effect of price-quality schedule changes on the optimal input choice of the firm. These stochastic dominance results are then extended by relaxing some dominance constraints CONSTRAINTS - A language for solving constraints using value inference. ["CONSTRAINTS: A Language for Expressing Almost-Hierarchical Descriptions", G.J. Sussman et al, Artif Intell 14(1):1-39 (Aug 1980)]. that have little meaning in the context of price schedule dominance. Then the theory of monotonic functions “Monotonic” redirects here. For other uses, see Monotone. In mathematics, a monotonic function (or monotone function) is a function which preserves the given order. [2, 13; 7, 197] is applied to study the effect of a global change from a constant price contract to a price-quality schedule contract. We also consider in detail the impacts of the commonly used minimum quality standard price-quality schedule on production. The paper is then summarized and conclusions are drawn. II. Marginal Changes in the Price Schedule Consider a producer facing a production function, F([Alpha], z), increasing and concave Concave Property that a curve is below a straight line connecting two end points. If the curve falls above the straight line, it is called convex. in [alpha]. Here, [alpha] is an input level (decision variable) and z is a quality index. Let the support of z be [a, b]. Then the producer's total output, over all z, is given by [MATHEMATICAL EXPRESSION A group of characters or symbols representing a quantity or an operation. See arithmetic expression. NOT REPRODUCIBLE re·pro·duce v. re·pro·duced, re·pro·duc·ing, re·pro·duc·es v.tr. 1. To produce a counterpart, image, or copy of. 2. Biology To generate (offspring) by sexual or asexual means. IN ASCII ASCII or American Standard Code for Information Interchange, a set of codes used to represent letters, numbers, a few symbols, and control characters. Originally designed for teletype operations, it has found wide application in computers. ] Let the price of [Alpha] be w, and the output price schedule faced by the producer be p(z), increasing in z. Therefore, the producer's profit is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] It is obvious that a new price schedule that exceeds (is exceeded by) the old price schedule over all but sets of measure zero on [a, b] will increase (decrease) the use of [Alpha]. That is, [Alpha] is monotonic monotonic - In domain theory, a function f : D -> C is monotonic (or monotone) if for all x,y in D, x <= y => f(x) <= f(y). ("<=" is written in LaTeX as \sqsubseteq). in p(z). It can be easily shown that if p(z) increases over a set of positive measure and does not decrease over any set of positive measure, then use of a increases. We will concentrate on price schedule changes where the new and the old schedules cross, because in these cases the production effect is not clear. To initiate the analysis, we assume that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where k is a constant. This assumption will be relaxed after the model has been established. To make progress, it is necessary to impose regularity conditions on F([Alpha], z). To ensure a unique solution to the maximization of equation (2) over [Alpha], we impose monotonicity of F([Alpha], z) in z. While either positive or negative monotonicity is plausible, we impose [F.sub.z] [greater than] 0, where the subscript (1) In word processing and scientific notation, a digit or symbol that appears below the line; for example, H2O, the symbol for water. Contrast with superscript. (2) In programming, a method for referencing data in a table. denotes a partial derivative partial derivative In differential calculus, the derivative of a function of several variables with respect to change in just one of its variables. Partial derivatives are useful in analyzing surfaces for maximum and minimum points and give rise to partial differential . For the theorems This is a list of theorems, by Wikipedia page. See also
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] We now present the first theorem theorem, in mathematics and logic, statement in words or symbols that can be established by means of deductive logic; it differs from an axiom in that a proof is required for its acceptance. : THEOREM 1. Consider any two price schedules, [p.sup.1](z) and [p.sup.2] (z), satisfying T1 a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.2] (z))] dz = 0, T1 b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.1](z)]dz [less than or equal to] - 0 for all y in [a, b], and the inequality inequality, in mathematics, statement that a mathematical expression is less than or greater than some other expression; an inequality is not as specific as an equation, but it does contain information about the expressions involved. is strict for at least one value of y. Then optimal [Alpha] increases (decreases) on the shift from the schedule [p.sup.1] (z) to the new schedule [p.sup.2](Z) [equivalence] [F.sub.[Alpha]z] [greater than or equal to] [(less than or equal to)] 0 for all z in [a, b] and for all feasible [Alpha]. Proof. Let the solution to the first order condition when schedule [p.sup.1] (z) pertains be [Alpha.sup.1]. Evaluate the first order condition for schedule [p.sup.2](Z) at the [Alpha] level of [Alpha.sup.1]. This is not necessarily equal to zero. Because [F.sub.Alpha] is monotonic decreasing in [Alpha], optimum [Alpha] increases (decreases) according as ac·cord·ing as conj. 1. Corresponding to the way in which; precisely as. 2. Depending on whether; if. [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - w [greater than] [(less than)] 0, where [|.sub.Alpha]=[Alpha.sup.1] means evaluated at [Alpha] = [Alpha.sup.1]. Subtract A relational DBMS operation that generates a third file from all the records in one file that are not in a second file. the first order condition pertaining per·tain intr.v. per·tained, per·tain·ing, per·tains 1. To have reference; relate: evidence that pertains to the accident. 2. when the price schedule is [p.sup.1](z) to find that [Alpha] increases (decreases) with the change from [p.sup.1](z) to [p.sup.2](Z) according as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [greater than] [(less than))] 0. Integrating by parts, this condition can be shown to be equivalent to [[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The inner integration is positive, and so the theorem's forward implication is proved. To prove that the sign condition on [F.sub.[Alpha]z] is necessary for the theorem to hold over all [Alpha] and all z in [a, b], let [F.sub.[Alpha]z] have the opposite sign over a support of arbitrarily small but positive measure. Then, as the magnitude of the [F.sub.[Alpha]z] having opposite sign can be arbitrarily large In mathematics, the phrase arbitrarily large is used in statements such as:
This proof is an adaptation of a first degree stochastic dominance proof provided by Diamond and Stiglitz [4, 340]. Theorem 1 can be directly related to first order stochastic dominance arguments. In the theory of choice under uncertainty, T1 a) would ensure that the probability weights sum to the same value, while T1 b) would ensure that schedule [p.sup.2](z) would dominate schedule [p.sup.1](z). The result states that the change in input use in response to a shift in the output price schedule depends on how marginal physical product changes with quality. Next we relax T1 a) to get Corollary corollary: see theorem. 1. Consider any three price schedules, [p.sup.1](z), [p.sup.2](z) and [p.sup.3] (z), satisfying C1 a) [p.sup.3](z) = [p.sup.2](z) + g(z), C1 b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[p.sup.2](z) - [p.sup.1](z)]dz = 0, C1 c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] [[p.sup.2](z) - [p.sup.1](z)]dz [less than or equal to] 0 for all y in [a, b). Then optimal [Alpha] increases on the shift from price schedule [p.sup.1](z) to the new price schedule [p.sup.3] (z) if [F.sub.[Alpha]z] [greater than or equal to] 0 and g(z) [greater than] 0 both hold for all z in [a, b]. Conversely con·verse 1 intr.v. con·versed, con·vers·ing, con·vers·es 1. To engage in a spoken exchange of thoughts, ideas, or feelings; talk. See Synonyms at speak. 2. , if g(z) [less than] 0 over [a, b] then optimal [Alpha] decreases on the shift from the old to the new schedule if [F.sub.[Alpha]z] [less than or equal to] 0 for all z in [a, b] and for all feasible [Alpha]. Proof. Apply Theorem 1 and monotonicity of a in output price. Q.E.D. Theorem 1, together with its corollary, fall well short of ranking all price schedules. T1 b) requires that the functions [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] never cross as y changes. Its corollary, permits such crossings. However, though extending the set of ranked schedule pairs considerably, the corollary still does not rank the universe of monotonic increasing functions (Math.) a function whose value increases when that of the variable increases, and decreases when the latter is diminished; also called a monotonically increasing function ltname>. See also: Increase . To this end, we consider in Theorem 2 a more general classification of price-quality schedules. THEOREM 2. Consider any two price schedules, [p.sup.1](z) and [p.sup.2](z), satisfying T2 a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.1](z)] dz = 0, T2 b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.1](s)]dsdy [less than or equal to] 0 for all z in [a, b], and the inequality is strictfor at least one value of z, T2 c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.1](s)]dsdy = 0. Then optimal [Alpha] increases (decreases) on the shift from price schedule [p.sup.1] (z) to the new schedule [p.sup.2](z) [equivalence] [F.sub.[Alpha]zz] [less than or equal to])[(greater than or equal to] 0 for all z in [a, b] and for all feasible [Alpha]. Proof: Applying the analysis in Theorem 1 we can no longer sign the expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] because T2 a) and T2 b) do not lend themselves to signing the inner integration. However, integrating by parts again, and using T2 c), we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From T2 b), the inner double integration is negative, and so the theorem's forward implication is proved. To prove that global uniformity of the sign of [F.sub.[Alpha]zz] is necessary for the theorem to hold over all feasible [Alpha] and all z in [a, b], let [F.sub.[alpha]zz], have the opposite sign over a support of arbitrarily small but positive measure. Then, as the magnitude of the [F.sub.[Alpha]zz] having opposite sign can be arbitrarily large, we cannot sign the double integration. This establishes necessity. Q.E.D. This theorem can be directly related to mean preserving contraction contraction, in physics contraction, in physics: see expansion. contraction, in grammar contraction, in writing: see abbreviation. contraction - reduction arguments [9, 227]. In this context T2 a) places equality on the sum of the probability weightings, while T2 b) and T2 c) are the conditions for a mean preserving contraction. We may interpret the theorem as stating that, under a change in the price schedule that satisfies the conditions, the optimal input choice depends only on the curvature curvature Measure of the rate of change of direction of a curved line or surface at any point. In general, it is the reciprocal of the radius of the circle or sphere of best fit to the curve or surface at that point. , with respect to quality, of the marginal product In economics, the marginal product or marginal physical product is the extra output produced by one more unit of an input (for instance, the difference in output when a firm's labour is increased from five to six units). curve. As there appears to be no economic reason for assigning as·sign tr.v. as·signed, as·sign·ing, as·signs 1. To set apart for a particular purpose; designate: assigned a day for the inspection. 2. a particular sign to this third derivative derivative: see calculus. derivative In mathematics, a fundamental concept of differential calculus representing the instantaneous rate of change of a function. , the production implication of shifting from [p.sup.1](z) to [p.sup.2](z) is not clear. As in Theorem 1, we now relax T2 a) to get Corollary 2. Consider any three price schedules, [p.sup.1](z), [p.sup.2](z) and [p.sup.3](z), satisfying C2 a) [p.sup.3](z) = [p.sup.2](z) + g(z), C2 b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.2](z)]dz = 0, C2 c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.1](s)]dsdy [less than or equal to] 0 for all z in [a, b], and the inequality is strict for at least one value of z, C2 d) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.1](s)]dsdy = 0. Then optimal [Alpha] increases on the shift from the price schedule [p.sup.1](z) to the new schedule [p.sup.[(z) if [F.sub.[Alpha]zz] [less than or equal to] 0 and g(z) > 0 both hold for all z in [a, b]. Conversely, if g(z) < 0 over [a, b], then optimal [alpha] decreases on the shift from [p.sup.1] (z) to [p.sup.3](z) if [F.sub.[Alpha]zz] 0 for all z in [a, b] and for all feasible [Alpha]. Proof: Apply Theorem 2 and monotonicity of [Alpha] in output price. Q.E.D. Just as corollary 1 extended considerably the ranked set of schedules, this corollary extends considerably the ability of theorem 2 to rank price schedules. It is worth noting that a pair of schedules that satisfy the conditions of theorem 1 do not satisfy the conditions of theorem 2, and vice versa VICE VERSA. On the contrary; on opposite sides. ; the conditions are mutually excluding. However, a pair of schedules may satisfy both of the corollaries. Applying the stochastic dominance analogy analogy, in biology, the similarities in function, but differences in evolutionary origin, of body structures in different organisms. For example, the wing of a bird is analogous to the wing of an insect, since both are used for flight. , the conditions laid out in theorems 1 and 2 are both special cases of the conditions for second degree stochastic dominance. Our next theorem identifies conditions for ranking price schedules that would, in the choice under uncertainty literature, be considered ranked by second degree stochastic dominance. THEOREM 3. Consider any two price schedules, [p.sup.1](z) and [p.sup.2], satisfying T3 a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.2](z)]dz = 0. T3 b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] - [p.sup.1](s)]dsdy [less than or equal to] 0 for all z in [a, b), and the inequality is strict for at least one value of z. Then optimal a increases (decreases) on the shift from the price schedule [p.sup.1] (z) to the new schedule [p.sup.2](z) if [F.sub.[Alpha] [less than or equal to] [(greater than or equal to] and [F.sub.[Alpha]z] 0 for all z in [a, b] and for all feasible Proof Applying the analysis in Theorem 1, we can no longer sign the expression [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Integrating by parts again, we get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] From [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is. Therefore the first integration (6) has the sign of [F.sub.Alpha.z]. Also from T3 b), expression (5) has the negative of the sip of [F.sub.Alpha.zz]. Therefore if [F.sub.Alpha.z] and [F.sub.Alpha.zz] have opposite signs, then the input effect can be signed using the second order condition [F.sub.Alpha Alpha] [is less than] 0. This establishes sufficiency. To show that both conditions are necessary if the theorem is to hold over all [Alpha] and all z in [a, b], let [F.sub.Alpha.z] and [F.sub.Alpha.zz] have the same signs over a support of arbitrarily small but positive measure. Then, as the absolute magnitude absolute magnitude: see magnitude. of either [F.sub.Alpha.z] and [F.sub.Alpha.zz] can be arbitrarily large, we cannot sign the double integration. This establishes necessity. Therefore, it is both necessary and sufficient to impose on the production function the conditions imposed in Theorems 1 and 2. Q.E.D. Theorem 3 relaxes the criteria set in Theorem 2 by removing its T2 c). It relaxes the criteria set in Theorem 1 by requiring that T1 b) apply cumulatively rather than for all z. Now, just as in Theorems 1 and 2 above, we relax T1 a), the constant weighting criterion, Corollary 3. Consider any three price schedules, [p.sup.1](z), [p.sup.2](z) and [p.sup.3](z), satifying C3 a) [p.sup.3](z) = [p.sup.2](z) + g(z), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] C3 c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], for all z in [a, b] and the inequality is strict for at least one value of z. Then optimal [Alpha] increases on the shift from the schedule [p.sup.1](z) to the new schedule [p.sup.3](3) if [F.sub.Alpha.zz] [equal to or less than] 0 and [F.sub.Alpha.z] [equal to or greater than] 0 and g(z) [is greater than] 0 hold for all z in [a, b] and all feasible [Alpha]. Conversely, if g(z) [is less than] 0 over [a, b], then optimal [Alpha] decreases on the shift from schedule [p.sup.1](z) to [p.sup.3](z) if [F.sub.Alpha.zz] [equal to or greater than] 0 and [F.sub.Alpha.z] [equal to or less than] 0 for all z in [a, b] and all feasible [Alpha]. Proof. Apply Theorem 3 and monotonicity of a in output price. Q.E.D. Theorem 3 and its corollary place the least restrictions on price schedule shifts, but at the cost of placing the most restrictive conditions on the production function. III. Global Changes in the Price Schedule Until now, we have considered price schedule changes which, in the choice under uncertainty literature, have been called marginal impact changes [10, 67). This is distinct from global, or overall, impact changes which arise when moving from a certain to an uncertain situation. From this literature, it is clear that the marginal and global impacts of uncertainty on optimal choice are quite distinct [5; 10]. In our next theorem, we will study the effect on optimal input choice of a global price schedule change, i.e. from a constant price to a quality dependent price schedule. We will first define the concept of increasing in mean [2, 13]. The function h(z) is increasing in mean with respect to the uniform (i.e., constant density) measure over [a, b] if the function [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is monotonic increasing in z. Differentiating with respect to z, we find that this is true if h(z) [is greater than] [Phi](z) for all z in [a, b]. Using the concept of increasing in mean we state THEOREM 4. Consider a price schedule, [p.sup.2](2), and a price, [p.sup.1], invariant (programming) invariant - A rule, such as the ordering of an ordered list or heap, that applies throughout the life of a data structure or procedure. Each change to the data structure must maintain the correctness of the invariant. to z. Let [p.sup.1] [equal to or less than] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Let [p.sup.2](2) and [F.sub.Alpha] be integrable, in the Lebesgue sense, in z over [a, b]. Let [p.sub.2](z) and [F.sub.Alpha] also be non-negative and bounded on [a, b]. Then the optimum choice of [Alpha] is higher under schedule [p.sup.2](z) than under [p.sup.1] if both [p.sup.2](z) and [F.sub.Alpha] are increasing in mean with respect to z. Conversely, if [p.sup.1] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is increasing in mean but [F.sub.Alpha] is decreasing in mean with respect to z, then the optimum choice of [Alpha] is lower under schedule [p.sup.2](z) than under [p.sup.1] Proof Define the functions [Phi]p(z) and [Psi]p(z) on (a, b] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Similarly, define the functions [Phi] [F.sub.Alpha] (z; a) and [Psi] [F.sub.Alpha], (z; a) on [a, b] by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [F.sub.Alpha] (z; a) is the marginal product of F ([Alpha], z) given [Alpha]. Let the solution to the first order condition, equation (4), when [p.sup.1] pertains be [Alpha.sup.1]. Evaluate the first order condition for schedule [p.sup.2](z) at the a level of [Alpha.sup.1]. This is not necessarily equal to zero. Because [F.sub.Alpha] is monotonic decreasing in [Alpha], optimum a increases (decreases) with the change from [p.sup.1] to [p.sup.2(z) according as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, [Alpha] increases (decreases) according as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Consider first the case where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] a). Therefore, [Alpha] increases if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] is positive. Next we define [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Differentiating with respect to z, we find that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Because both [p.sup.2](z) and [F.sub.Alpha], are increasing in mean, we use (7), (8), and (9) to find that the last right-hand side right-hand side n → derecha right-hand side right n → rechte Seite f right-hand side n → lato destro of (12) is positive for all z in [a, b]. Integrating over [a, b], we find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] But, from evaluating (11) at z = b, this is (10), the condition under which a is greater with schedule [p.sup.2](z) than with a constant price equal to [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the inequality remains true for the quality invariant price schedule [p.sup.1]. The converse (logic) converse - The truth of a proposition of the form A => B and its converse B => A are shown in the following truth table: A B | A => B B => A ------+---------------- f f | t t f t | t f t f | f t t t | t t case is proved in a similar manner. Q.E.D. This theorem differs from Theorem 1 and its corollary in that no sign restriction is placed on [F.sub.Alpha.z]. In fact, while [F.sub.Alpha], must trend upwards (or, conversely, downwards) in z, it may change sign an indefinite number indefinite number n. A variable number. of times. The result has been strengthened, relative to Theorem 1, for two reasons: First, because we are considering a global, rather than a marginal, change in the price schedule; second, because we use a property of the price schedule not previously used, the assumed general tendency of price to increase with quality. IV. Minimum Quality Standards Having considered price schedules from a stochastic dominance approach, we will now accommodate minimum quality standards, a common feature of production contracts. This can be viewed as just a particular form of price schedule. Here, we drop the constraint Constraint A restriction on the natural degrees of freedom of a system. If n and m are the numbers of the natural and actual degrees of freedom, the difference n - m is the number of constraints. concerning the monotonicity of output in quality. First we prove Theorem 5. If a minimum quality standard, [z.sub.A], is in place, and if all products with quality higher than [z.sub.A] receive a common, quality standard related price, p([z.sub.a]), then [Alpha] increases (decreases) with [z.sub.A] according as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where Ln is the natural log operator Proof Producer profit is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The first order condition is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Differentiating totally with respect to a and [z.sub.A], we find [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This is positive (negative) if the numerator numerator the upper part of a fraction. numerator relationship see additive genetic relationship. numerator Epidemiology The upper part of a fraction is negative (positive). The numerator can be written more succinctly suc·cinct adj. suc·cinct·er, suc·cinct·est 1. Characterized by clear, precise expression in few words; concise and terse: a succinct reply; a succinct style. 2. as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Because [F.sub.Alpha] and P([z.sub.A]) are strictly positive, the sign is determined by the term in large brackets brackets: see punctuation. . Q.E.D. Because output is monotonic increasing in a, this result can be interpreted as follows. It is the requirement that if an increase in the quality standard is to increase total output, then price must increase at a rate faster than the rate at which the marginal product of marketed output decreases with [z.sub.A]. Price will increase because the quantity of marketed output may decrease, and because the average quality of marketed output increases. Having identified the price-minimum quality standard relationship condition under which production will increase with an increase in the minimum quality standard, the question naturally arises; under what condition will total marketed output increase? Denote de·note tr.v. de·not·ed, de·not·ing, de·notes 1. To mark; indicate: a frown that denoted increasing impatience. 2. total marketed output by G, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] The total effect of [z.sub.A] on marketed output equals the direct effect added to the indirect effect through altering the use of [Alpha]; [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] Substituting in (17) and (18), and isolating i·so·late tr.v. i·so·lat·ed, i·so·lat·ing, i·so·lates 1. To set apart or cut off from others. 2. To place in quarantine. 3. the price effects, we find that marketed output will increase (decrease) according as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The first term in the product on the right is the negative of the tradeoff between [Alpha] and [z.sub.A] for a given level of saleable sale·a·ble adj. Variant of salable. saleable or US salable Adjective fit for selling or capable of being sold saleability or US output, and is negative. The second term is the proportional proportional values expressed as a proportion of the total number of values in a series. proportional dwarf the patient is a miniature without disproportionate reductions or enlargements of body parts. response of the tradeoff to [Alpha], and can be shown to be negative. Condition (20) may be re-written as [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] To find the difference between critical rates (20) and (14), subtract the right-hand side of (14) from the right-hand side of (21) to get [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] This is the difference between the rate of price increase that maintains total marketed production and the rate of price increase that maintains total production in response to an increase in the minimum standard. It is positive because total marketed output is both concave in a and decreasing in the minimum quality standard. The first bracketed term is the tradeoff between [z.sub.A] and a that maintains total marketed output. If the absolute value of the tradeoff is high, then the gap between the price incentive that maintains total marketed output and the price incentive that maintains total output must be large enough to induce in·duce v. 1. To bring about or stimulate the occurrence of something, such as labor. 2. To initiate or increase the production of an enzyme or other protein at the level of genetic transcription. 3. the profit maximizing firm to use more [Alpha]. The second bracketed term is a measure of the curvature of marketed output in a. If the absolute value of the curvature is high, then the price gap must be large. V. Summary and Conclusions The purpose of this paper has been to specify and analyze the microeconomic mi·cro·ec·o·nom·ics n. (used with a sing. verb) The study of the operations of the components of a national economy, such as individual firms, households, and consumers. production implications of a change in the price-quality schedule for the class of production technologies where producers have some, but not complete, control over the quality of output produced. We use a very general production technology, and employ stochastic dominance methods to rank price-quality schedules according to the input usage levels that they induce. We progress successively from applying restrictive price schedule dominance criteria and unrestrictive Adj. 1. unrestrictive - not tending to restrict restrictive - serving to restrict; "teenagers eager to escape restrictive home environments" production technology assumptions to applying less restrictive dominance criteria and more restrictive production assumptions. The slope and curvature of marginal product with respect to quality are found to be important in signing comparative static effects. We also consider the global impact of moving from a quality invariant price schedule to a quality dependent price schedule. Here, conditions less restrictive than monotonicity of price and marginal product, with respect to quality, are sufficient to sign the impact. Finally, we investigate the relationships between the production technology, a minimum quality standard and the price levels required to maintain total and marketed output. It is found that for total output to increase with the minimum quality standard, the price of output must increase at a rate faster than the rate at which marginal product of marketed output decreases with the standard. For total marketed output to increase, the percent price increase must of course be greater still, and the magnitude of the difference depends on the slope and curvature of the production function. As the issues raised are important and as relatively little work has been done in the area, there appears to be much potential for further research. In particular, the techniques employed in this paper may provide a theoretical framework for analyzing pricing contracts, grading structures, quality standards, and agricultural marketing orders. The paper also highlights the need for research on a complete classification of price schedules according to their production implications. References [1.] Bowbrick, Peter. The Economics of Quality, Grades and Brands. Routledge For people named Routledge, see . Routledge is a publisher of non-fiction academic books. It was acquired in 1997 by, and is thus now an imprint of the Taylor & Francis Group, which is a sub-division of Informa PLC, a company based in the United Kingdom with offices worldwide. , Chapman and Hall Chapman and Hall was a British publishing house, founded in the first half of the 19th century by Edward Chapman and William Hall. Upon Hall's death in 1847, Chapman's cousin Frederic Chapman became partner in the company, of which he became sole manager upon the retirement of , Inc., New York New York, state, United States New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : 1992. [2.] Burkill, H. and L. Mirsky, "Comments on Chebycheff's Inequality." Periodica Mathematica Mathematical software for the Macintosh, DOS, Windows, OS/2 and various Unix platforms from Wolfram Research, Inc., Champaign, IL (www.wolfram.com). Launched in 1988, Mathematica includes numerical, graphical and symbolic computation capabilities, all linked to the Mathematica programming Hungarica, 6(1) 1975, 3-16. [3.] Choi Choi may refer to:
[4.] Diamond, Peter A. and Joseph Stiglitz, "Increases in Risk and Risk Aversion risk aversion The tendency of investors to avoid risky investments. Thus, if two investments offer the same expected yield but have different risk characteristics, investors will choose the one with the lowest variability in returns. ." Journal of Economic Theory, July 1974, 337-60. [5.] Meyer, Jack and Michael B. Ormiston, "Strong Increases in Risk and their Comparative Statics Comparative statics is the comparison of two different equilibrium states, before and after a change in some underlying exogenous parameter. As a study of statics it compares two different unchanging points, after they have changed. ." International Economic Review, June 1985, 153-69. [6.] Ormiston, Michael B. and Edward E. Schlee, "Necessary Conditions for Comparative Statics under Uncertainty." Economics Letters Economics Letters is a scholarly peer-reviewed journal of economics that publishes concise communications (letters) that provide a means of rapid and efficient dissemination of new results, models and methods in all fields of economic research. Published by Elsevier. , December 1992, 429-34. [7.] Pecaric, Josip E., Frank Proschan, and Y. L. Tong tong 1 tr.v. tonged, tong·ing, tongs To seize, hold, or manipulate with tongs. [Back-formation from tongs. . Convex Functions In mathematics, a real-valued function f defined on an interval (or on any convex subset of some vector space) is called convex, or concave up, if for any two points x and y in its domain C and any t in [0,1], we have [8.] Rosen, Sherwin, "Hedonic he·don·ic adj. 1. Of, relating to, or marked by pleasure. 2. Of or relating to hedonism or hedonists. [Greek h Prices and Implicit Markets: Product Differentiation Product Differentiation A source of competitive advantage that depends on producing some item that is regarded to have unique and valuable characteristics. in Pure Competition." Journal of Political Economy, January-February 1974, 34-54. [9.] Rothschild, Michael and Joseph Stiglitz, "Increasing Risk: I. A Definition.- Journal of Economic Theory, September 1970, 225-43. [10.] Sandmo, Agnar, "On the Theory of the Competitive Firm under Price Uncertainty." American Economic Review, March 1971, 65-73. [11.] Wright, Brian D., "The Effects of Price Uncertainty on the Factor Choices of the Competitive Firm." Southern Economic Journal, October 1984, |
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