Quality Versus Quantity in Vertically Differentiated Products Under Non-Linear Pricing

Quality is defined as being skewed when the marginal rate of substitution (MRS) between quantity and quality differs from the marginal rate of transformation (MRT). This definition is used to assess the balance of quality and quantity in each variety of good produced by a monopolist using non-linear pricing, where each variety can be differentiated using both quantity and quality. A variety’s decisive customers face a binding self-selection or participation constraint. Skewing of a variety’s quality occurs when there is a difference between its decisive customer(s) MRS and that of (i) its non-decisive customers (ii) the decisive customers of ‘adjacent varieties’. Some important special cases are identified and analysed. JEL Classification: L11 ∗ This paper utilises and extends some of the analysis of Sibly (2006). I would like to that Jong-Hee Hahn and Ann Marsden for comments on an earlier draft. All errors remain my responsibility. QUALITY VERSUS QUANTITY IN VERTICALLY DIFFERENTIATED PRODUCTS UNDER NON-LINEAR PRICING One often hears people make statements such as “I am a person who prefers quality over quantity”. The implications of such comments are, of course, that such people recognize there is a trade-off between these attributes. Unlike this popular saying, people do not have a preference for either quantity or quality but, rather, have a rate of substitution between the two. Such preferences will be apparent in the individuals’ demand, and therefore market equilibrium. When customers differ in their preferences of quality, it is in the interest of the firm to vertically differentiate its product (Mussa and Rosen, 1978). When customers differ in their preferences for quantity and quantity it will be in the interests of firms to differentiate its products which so that bundles differ in both quantity and quality. For example, suppliers of instant coffee usually provide their high quality variety in relatively small jars, whereas the low quality variety brand is supplied in larger jars. In contrast, some products that could be readily vertically differentiated, for example rice, are only supplied in varying quantities but only in one quality. The aim of this paper is to provide an economic interpretation of the equilibrium in these, and analogous, examples. A definition of “skewed quality” is used to assess the balance of quality and quantity in each of the (equilibrium) bundles supplied by the firm. Quality is said to be skewed if the marginal rate of substitution (MRS) between quantity and quality differs from the marginal rate of transformation (MRT). This method of analysis differs from traditional practice in that it looks at the balance between quantity and quality rather than looking at the impact of marginal increments in quantity and quality individually. For example, an equilibrium variety (jar) of coffee would have skewed quality if, for a given level of cost, a substitution of quantity for quality results in a higher consumer benefit (and hence social surplus). This paper provides an analysis that allows identification of the skewness of quality of varieties in a market with a wide class of benefit and cost functions, To place the contribution of this paper in context it is useful to review the existing literature. The pioneering papers of Spence (1975) and Sheshinski (1976) modelled the quality of a single variety supplied by a monopolist who utilizes linear pricing. These papers implicitly incorporated a trade-off between quantity and quality in a general way. These papers showed (following Swan, 1970) that there was no unambiguous relationship between the efficient and profit maximizing quality level. With Mussa and Rosen (1978) interest turned to the role of quality in differentiating products. Mussa and Rosen (1978), and the overwhelming bulk of the subsequent literature adopted the unit demand model. In these models each customer has a unit demand, and increases in quality linearly increase the willingness to pay. As is extremely well known, Mussa and Rosen show that all but the highest quality good is supplied at a sub-optimal quality. The customers purchasing the low quality good receive no consumer surplus, however all other customers do. These results rely critically on Mussa and Rosen’s use of the unit demand model and of the ordering of utility in that model. In subsequent work this approach provides great analytical convenience, and it has often proved possible to provide unambiguous results that relate quality to its efficient level. However the unit demand model does not allow substitution possibilities between quantity and quality. It thus abstracts from one of the fundamental features of markets with endogenous quality. In addition, the unit demand model blurs the distinction between linear and nonlinear pricing. Maskin and Riley (1984) consider how a monopolist can use nonlinear pricing to conduct price discrimination. In equilibrium the monopolist bundles output: low valuation customers purchase bundles with an inefficiently low quantity, whereas the highest valuation customer purchases a bundle with an efficient quantity. Maskin and Riley note that their model can be recast as the problem of vertical differentiation studied by Mussa and Rosen (1978). In doing so they adopt, as did Mussa and Rosen, the unit demand model. The results of this exercise parallel both their analysis of nonlinear pricing and Mussa and Rosen’s results. Specifically they show, assuming a restriction on consumer preferences known as the “single crossing property”, that the quality level supplied to low valuation customers is below the efficient level, whereas the quality level supplied to the highest valuation customer is efficient. The preferences assumed by Mussa and Rosen (1978) and Maskin and Riley (1984) are ones that can be ordered by a single parameter (characteristic). These analyses are thus referred to as uni-dimensional screening problem. There has been interest in extending these early results to cases involving more general preferences, particularly to preferences allowing substitution possibilities between ‘instruments’ of a firm’s output. This leads to a ‘multidimensional screening problem’, in which the aim is to identify the profit maximizing nonlinear pricing schedule given the assumed distribution of preferences and cost function. Unfortunately, multi-dimensional screening problems, particularly in case of non-linear pricing, turn out to be analytically challenging. Rochet and Choné (1998) provide a useful survey of this small, and technical, literature. They note that the literature on multidimensional screening can be delineated by the number of instruments available to the firm relative to the number of characteristics of customers. Laffont, Maskin and Rochet (1987) consider the case in which the firm has one instruments but faces customers with two characteristics. On the other hand, the firm may have multiple instruments but face customers who have only one characteristic (Mathews and Moore, 1987). Finally the both the firm and customers may both have multiple characteristics (Armstrong, 1996, Rochet and Choné, 1998 and Sibley, Srinagesh, 1997, Wilson, 1993). Armstrong (1996) shows, in the context of a multi-product monopolist, that there are some (low valuation) customer types who will not be supplied by the monopolist. Rochet and Choné (1998) consider a monopolist who supplies one unit to each customer, but each unit is differentiated by multiple quality characteristics. They use a generalization of the approach of Mussa and Rosen (1978) to show that bunching (i.e. selling the same good to customers with differing characteristics) will commonly occur in multidimensional settings. The purpose of this paper is not to provide a new solution to multidimensional screening problems, but to provide an economic interpretation of market equilibria in which the firm can differentiate bundles using both quantity and quality. (Hence the firm has two instruments.) The requirement to satisfy the self-selection constraints causes the equilibrium bundle of each variety to differ from the efficient one. The difference between each variety’s equilibrium bundle and its efficient one can be decomposed into an unskewed and a skewed component. Intuitively the unskewed component representing the distorting of the ‘desirability’ of the bundle (while maintaining the optimal balance of quality and quantity), the skewed component represents the distorting of the balance of quality and quantity. It is shown that in equilibrium the skewness of variety i depends on (i) the weighted differences in the marginal rates of substitution of the customer types purchasing variety i and (ii) the difference between the variety i’s customers marginal rate of substitution and the weighted marginal rate of substitution of other variety’s customers who would switch to variety i with a marginal lowering of its fee. When all customer types have a common marginal rate of substitution, no skewing of quality and quantity occurs. In this event, self-selection is contingent on the ‘desirability’ of 1 The firm considered in this paper is one that bundles quantity and quality. This problem is formally identical to the one facing a monopolist that utilizes non-linear pricing to sell bundles consisting of two goods. 2 In Rochet and Choné’s model the unit could be interpreted a single ‘bundle’ and one of the quality dimensions could be interpreted as quantity. the bundle, which can be ordered in one dimension. When, further, cost takes an iso-elastic functional form, the monopolist’s problem can be transformed into, what is effectively, a onedimensional problem. If additionally, in some case in which customer types’ preferences can be represented by an iso-elastic benefit function, either quantity or quality can be common across all bundles. These cases represent a link between the one-dimensional analyses of Mussa and Rosen and Maskin and Riley, and the multidimensional analyses discussed above and in this paper. Section 1 states the formal definition of skewed quality and discusses its implications. It also includes descriptions of equilibrium, including the distinction between bunched and unbunched equilibrium Section 2 states the monopolist’s optimization problem. Sections 3 considers the cases in which preferences are such that no variety exhibits skewed quality. Section 4 considers the skewing of quality that occurs in the relatively straightforward cases of unbunched equilibria. Section 5 considers the skewing that occurs in all categories of equilibrium that can occur when there are three customer types. Section 6 the skewing of quality that occurs when there are two varieties produces, and many customer types purchasing both varieties. Section 7 concludes the paper. 1. Equilibrium and Skewed Quality A monopolist produces n vertically differentiated varieties of good. Each variety i is a bundle with quantity X of quality Z. The firm offers a set of schedules Ω ≡ , i=1,..n, in which the bundle for fee T. The equilibrium (profit maximizing) schedule for variety i is denoted Ω* i ≡. A monopoly has a number of customer types that differ in preferences. For convenience the types are ordered by two indexes, i and φ with φ=1,..ni. This allocation of indexes is chosen in such a way that in equilibrium customers types iφ purchase variety i. The total number of customer types is thus ∑ i=1 n ni. There are N type iφ customers. Each variety has N= ∑ φ=1 ni N customers in equilibrium, and the total number of customers is N= ∑ φ=1 n N. Note that much of the literature discussed in the introduction assumes a continuum of customer types. However, while this assumption aids in the derivation of solutions for a large number of customer types, the assumption of a finite number of customer types is more appropriate for the economic interpretation presented in this paper. The consumer benefit (utility) type iφ customers receive from consuming variety i is V(X,Z). The surplus from production of all varieties is therefore: S(X,Z,X,Z) = ∑ i=1 n [NV(X,Z)-C(NX,Z)] (1) where V(X,Z) = ∑ φ=1 ni (N/N)V(X,Z) is the average consumer benefit of the customers of purchasing variety i and C(NX,Z) is the total cost of producing variety i. It is assumed that cost is non-decreasing in quantity and quality, C1(NX,Z)>0, C2(NX,Z) >0, C11(NX,Z)≥ 0 and C22(NX,Z)≥0. In contrast to much of the literature there may be a fixed cost of producing each variety, so C(0,Z)≥0. The following definition is useful to discuss the importance of this latter assumption. Following Mussa and Rosen (1978) and Rochet and Choné (1998): Definition 1: An equilibrium schedule, Ω* , is unbunched if ni=1 and bunched if ni>1. An equilibrium {Ω* , i=1,...,n} is unbunched if ni=1 for all i =1,..n, and bunched if ni>1 for at least one i. The literature usually restricts consideration to those cases in which bunching would not occur under first-degree price discrimination (in which firms can costlessly identify the type of a particular customer). In such cases each customer type is provided with a unique bundle (with the fee equal to their consumer benefit). This outcome usually requires the adoption of the constraint returns to scale cost function. In such instances, bunching emerges as the firm’s optimal response to the need to satisfy the self-selection constraints. However bunching may occur for cost as well as screening reasons. In particular the firm may choose to adopt bunched schedules when there are fixed costs in offering additional varieties. The analysis of this paper allows for bunching for either cost or screening reasons. The following definition introduces the method that is used in this paper to evaluate equilibrium: Definition 2: In an equilibrium {Ω* , i=1,...,n}, the quality of variety i is said to be downwardly (un-,upwardly) skewed if: V 1( X * , Z* ) V 2( X * , Z* ) < (=, >) NC1(NX * , Z* ) C2(NX * , Z* ) (2) where the LHS is the marginal rate of substitution (MRS) and the RHS is the marginal rate of transformation (MRT) of variety i. The motivation for this definition is the observation that the unskewed combination of quantity and quality maximizes the surplus for a given level of resources devoted to production. Specifically the unskewed bundles of quantity and quality, denoted {X i,Z i} satisfies: {X i(C _ ),Z iC _ ))} = X Z argmax S(X,Z,X,Z) subject to C(NX,Z) ≤ C _ (3) where C _ is a given level of cost. The unskewed combination has optimal balance of quantity and quality. A variety has downwardly skewed quality if, by substituting quality for quantity, a higher surplus is obtainable for a given level of resources devoted to production of that variety. This is depicted graphically in figure 1 for variety i. Quality is unskewed at the point Π ~ i where the indifference curve V ~i is tangent to C* , the iso-cost curve. That is, quality is unskewed when the quantity-quality combination is {X i, Z i}. Note that at the point Π* i quality is downwardly skewed, as the MRS) X̂ (4) and the quality of variety i is said to be downwardly (un-,upwardly) distorted if: Z* i < (=, >) Ẑ (5) where X̂ is the efficient level of quantity and Ẑ is the efficient level of quality of variety i, defined by: V 1(X̂ , Ẑ) = C1(NX̂, Ẑ) and V 2( X̂ , Ẑ) = C2(NX̂, Ẑ) (6) Definition 3 is the widely adopted as the definition of quality distortion, including in the seminal work of Spence (1975) and Sheshinski (1976). This condition has proved a difficult analytical tool to use, (see, for example, Spence and Sheshinski). In addition, the interpretation of the ‘efficient level of quantity and quality of a variety’ is ambiguous when bunching occurs for screening purposes. In this case Pareto efficiency requires ‘unbunching’ of the variety, and offering each customer type their own specific variety. In contrast definition 3 describes the efficient bundle on the assumption that unbunching does not occur, and thus abstracts from issues associated with unbunching. Note also that definition 3 does not capture the idea of there being an optimal trade-off quantity and quality at different production levels Note that the implicit function theorem can be used to rewrite (3) to describe the unskewed bundles by the function Z = z ~i(Xi). The function z ~i is called the contract curve for variety i, as it represents the locus of points of tangency between the indifference curves and the iso-cost curves. Quality is skewed when the equilibrium bundle does not lie on the contract curve. In particular, the difference between the efficient and efficient and the equilibrium bundle, (X* -X̂, Z* -Ẑ), can be decomposed into two components: (i) the unskewed component is (X i-X̂i, Z i-Ẑi) and (ii) the skewed component is (X* -X i, Z* -Z i). 2. The Monopolist’s Optimisation Problem The consumer surplus of type iφ customers from the schedule is: U(X,Z,T) = ⎩⎪ ⎨ ⎪⎧V(X,Z) T if V(X,Z) ≥ T 0 if V(X,Z) < T (7) where it is assumed that customers who do not purchase a bundle receive zero benefit. The firm knows the distribution of customer types, but cannot identify specific individuals as belonging to a customer type. To ensure that type i customers purchase variety i the schedules must satisfy the selection constraints: V(X,Z) – T ≥ V(X,Z) – T for all φ=1,..ni and j≠i (8) Types iφ only purchase variety i if, in addition to the self selection constraints holding, the following participation constraint also holds: