Quality premiums and the firm's production decision.

• Author: Hennessy, David A.

1. 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 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 [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 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 that have little meaning in the context of price schedule dominance. Then the theory of monotonic functions [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.

2. Marginal Changes in the Price Schedule

   Consider a producer facing a production function, F([Alpha], z), increasing and concave 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 NOT REPRODUCIBLE IN ASCII]

   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 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 denotes a partial derivative. For the theorems presented below, modified results can be developed when F([Alpha], z) is monotonically decreasing in quality. In all cases, because output is monotonic in a, signing the effect of a price schedule change on a sips the effect on output. The first order condition for profit maximization is

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   We now present the first theorem:

   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 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 [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 the first order condition pertaining 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, and can dominate the whole expression, we can not sign the double integration. This establishes necessity. Q.E.D.

   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...