Conditions for Efficiency in Package Pricing


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1 Conditions for Efficiency in Package Pricing Babu Nahata Department of Economics University of Louisville Louisville, Kentucky 40292, USA. and Serguei Kokovin and Evgeny Zhelobodko Department of Economics, Novosibirsk State University Novosibirsk , Russia. [July 2004]
2 Conditions for Efficiency in Package Pricing Abstract: This paper shows that when a monopolist uses a nonlinear pricing (package pricing), and consumer valuations are nonordered (or demands cross), then the profitmaximizing output sold to all types of consumer can also be welfare maximizing. Conditions derived show that this result is not rare and some observable indicators can be used to infer about this overall socially efficient outcome. JEL Codes: D42, L10, L40
3 Conditions for Efficiency in Package Pricing 1. Introduction Under a monopoly, except under perfect discrimination, efficient or welfaremaximizing output is rather unlikely. For example, when a monopolist uses package pricing to serve two or more groups of consumers, the well known result in the literature is that the output sold only to one (highdemand) group is efficient. The output sold to lowdemand consumersislessthantheefficient output, and therefore there is some deadweight loss. Thecommonwisdom,whichisbasedonmostoften studied unidimensional and ordered cases, asserts that both groups cannot be served with sociallyefficient output. In contrast, in this note we state conditions when all groups of consumers can be served with efficient outputs, and moreover, such situations are rather realistic. In addition, we also show that when one observes tariffs paidbydifferent groups do not differ much, then it may indicate that all consumer groups are served with efficient outputs. In particular, when two groups pay the same tariff (or the same netofcosts tariff) fortwodifferent quantities, then it can be concluded that both quantities are efficient. Theoretical possibility of equal tariffs fordifferent (efficient) quantities provide a plausible explanation for some counterintuitive pricing practices regularly used by some airlines and softdrink sellers. Indeed, some airlines often times offer oneway (a smaller package in terms of miles flown) and roundtrip tickets (a larger package) at the same fare. In Japan two different sizes (300ml vs. 500ml) of CocaCola containers are regularly sold at the same price through vending machines, and in fact, different consumers do buy 2
4 both smaller and larger containers at the same price. These observed pricing practices as well as intellectual curiosity about how general the common wisdom is have provided the impetus for this note. We consider package pricing, which is a particular form of nonlinear pricing. 1 Consider a monopolist who uses a set of packages (quantitytariff bundles) as selfselection devices to identify hidden types of consumers. Packages are offered on takeitorleaveit basis, i.e., the quantity contained in each package is nonnegotiable. Because the monopolist, a priori, cannot identify the hidden types of consumers, he must take into consideration the socalled incentivecompatibility and participation constraints in designing the sizes and tariffs fordifferent packages. We consider package sizes and corresponding tariffs when consumervaluations are nonordered. The often considered case of ordered valuations means that the indifference curves of consumers cross only once, and satisfy the socalled singlecrossing property (or the SpenceMirrless condition). In terms of demand it is called the noncrossing condition, implying that the inverse demand functions do not cross. In reality the demand functions can intersect at some nonzero price when there is a consumer whose choking price (the willingnesstopay for an infinitesimally small unit) is relatively high and the demand curve declines faster than that of another consumers whose choking price is relatively smaller and the quantity that can be consumed at zero price (horizontal intercept) is bigger. 2 1 Broadly speaking, nonlinear pricing is a multidimensional screening problem. In recent years analyses of multidimensional screening problems have attracted considerable theoretical interest. For an excellent comprehensive review, see, Rochet and Stole (2003). 2 For example, demands would cross when a very thirsty consumer drinks smaller amount than a less thirsty consumer who can drink a lot. This seems to be the case between the business and leisure 3
5 2. Model Consider a case when a monopolist sells a homogeneous good using quantitytariff packages to two unobservable types of consumers, who selfselect the preferred package. Consumer cannot buy more than one package, and arbitrage is prevented. To simplify the analysis we assume that all consumers of a particulartypechoosethesamepackage.under this setting it is sufficient for the monopolist to design exactly two packages. Let the total cost function C(x) be linear so that the marginal cost c is a constant. Let the frequencies of two types be m 1 and m 2, respectively and their ratio is denoted by γ = m 2 /m 1. We assume that the utility functions are quasilinear of the form V i (x i ) T i,and depend on quantity x i 0 and tariff (outlay) T i R, sonoincomeeffects are present. Each valuation function V i (x i ) is concave and normalized as V i (0) = 0. Monopolist maximizes expected profit π with respect to variables (x, T )=(x 1,T 1,x 2,T 2 ). The standard formulation of the monopolist s optimization problem in this situation is: π(x, T )/m 1 = T 1 + γt 2 max x,t s.t. V i (x i ) T i V i (x j ) T j, V i (x i ) T i 0; x i 0 (i, j =1, 2) When any incentivecompatibility constraint like V i (x i ) T i V i (x j ) T j becomes becomes travellers. The choking price for business travellers is relatively high and satiation level of quantity small. For the leisure travellers, the opposite seems to be case the choking price is relatively low but their satiation level of quantity is relatively higher. Thus the inverse demand functions of these two types of travellers would intersect at some positive price. 4
6 an equality, we call it active and, following Wilson (1993), we interpret this situation as consumer i almostenvying consumer j s package. Our main goal is to show that social efficiency is tightly connected with the absence of any envy and that efficient outcomes may not be rare. Further, some observable indicators can indicate its possible presence in real life. 3. Main results We show conditions under which the solutions to the above program results in socially efficient outcome, i.e., the outcome that maximizes total welfare. The following proposition state the conditions, thereby generalizing the examples given in Rochet and Stole (2003). Proposition 1: For any package scheme ( x, T )tobeprofitmaximizing and socially efficient at the same time, it is both necessary and sufficient that the related valuation functions satisfy the following conditions. V i ( x j ) V j ( x j ) 0, i, j (incentivecompatibility), T i = V i ( x i ) (participation), x i =argmax x 0 (V i (x) C(x)) (welfare maximization). Proof: To prove this proposition, it is sufficient to note that these conditions actually imply that ( x, T ) is an unconstrained maximum of the total welfare function and that the entire consumer surplus is captured by the monopolist. Consider a graphical exposition of the situations in question to see that efficient outcomes do exist and are also probable. 5
7 [FIGURE 1] InthetopgraphinFigure1,eachpackage(x, T )representsapointinquantitytariff space. The valuation functions V 1, V 2 (for simplicity the curves) represent special indifference curves of utility functions, which are active at (0, 0) and the optimal point ( x, T ). Each consumer s curve includes all packages where her payoff isthesameaswhen she does not participate in trade. All packages below this curve (the lower, the better) would bring her some nonzero consumer surplus. For the monopolist the isoprofit lines (the higher, the better) are shown by lines parallel to the cost function cx going from the origin. The points of tangency ( x 1, T 1 ), ( x 2, T 2 )ofisoprofit lines with the valuation curves are the welfaremaximizing (efficient) packages. When each point of tangency happens to be above another consumer s curve, then this packageplan is incentivecompatible and profitmaximizing. The figure demonstrates that such occurrences are nondegenerate. Indeed, it is possible to somewhat modify valuations and costs and retain the same effect on efficiency. Note that because of numerous possibilities that exist for such modifications, efficient outcomes can occur with noticeable probability. To confirm this, fix thefirst curve in Figure 1. One can draw many different shapes of the second curve that would maintain efficiency. One can see that any concave curve whose tangency point ( x 2, T 2 ) lies within the shaded area suffices for the optimal packageplan to be welfaremaximizing, and that this area is not very small. 3 It is easy to visualize the above condition for efficiency packages in terms netofcosts 3 In contrast, the dotted line shows the necessary condition for efficiency, in the sense that when a concave valuation curve of the second consumer has tagency point above this dotted line, then efficiency is impossible. This observation shows that actual area of the possible efficiency region is bigger than the shaded area. 6
8 valuations (the bottom graph in Fig.1) and of demand functions (middle graph in Fig 1). Overall efficient outcome is quite probable when two demands cross and the two demand triangles above the marginal cost line are approximately equal in the areas. This property (considered in more detail below) is reflected by approximately equal heights of the peaks of netofcosts valuations on the bottom curve in Fig.1. Any couple of such curves results in efficient outcome and one can see a potential for continuum of such curves in the graph. Note that the above condition for efficiency does not depend on the relative frequency of the two types of consumers. Figure 1 can be used to explain the possibility of the same tariff for two different size packages. Suppose the demands are satiable and decline after the satiation level of quantity is reached (i.e., consumers derive disutility beyond the satiation level of consumption). It may be the case with the example of CocaCola containers mentioned earlier because consumers may find it inconvenient to carry a bigger bottle while walking on the street. Disutility may also result from an environmentally conscious consumer concerned with waste. Disutility may not be the reason with the roundtrip ticket example, but this may be a borderline case of disutility. Indeed, a passenger who wants to go only oneway does not get any additional utility from buying a roundtrip ticket, in spite of the fact that both prices are equal. In both these examples not only is the demand satiable, but also the cost difference between small and large package can be considered negligible compared to the fixed cost associated with any package. Without small differences between the cost, thesametariff for two different sized packages is hardly probable. We now turn to stating the conditions for efficiency in terms of variables that may be 7
9 observable (at least approximately) and can be used to infer the presence of the efficient outcomes in real life. In particular, we state the conditions in terms of package sizes, tariffs and costs. Proposition 2. Assume concave valuations. If both optimal packages (x 1,T 1 ) (x 2,T 2 ) have the same nettariffs in the sense that (T 2 T 1 )=c (x 2 x 1 ), then both are sociallyefficient. If (T 2 T 1 ) c (x 2 x 1 ) and the bigger package is efficient, then the smaller one is also efficient. Similarly, when (T 2 T 1 ) c (x 2 x 1 ) and the smaller package is efficient, then the bigger one must also be efficient. Proof. We use a direct graphical proof to prove the proposition. To prove the statement with inequality (T 2 T 1 ) c (x 2 x 1 ), consider Figure 1 and draw a line L 2,which is tangent to the curve V 2 at the point (x 2,T 2 ). Our condition implies that the smaller package (x 1,T 1 )cannotbebelowl 2 because the efficient package (x 2,T 2 )impliesthatl 2 is parallel to isoprofit lines. Suppose that the isoprofit line is not tangent to V 1 at the point (x 1,T 1 ) but intersects V 1. Then there are other points on V 1 that bring higher profit. They are as incentivecompatible as (x 1,T 1 ), taking into account the concavity of V 2, and line L 2 being tangent to V 2. The existence of such points contradicts the optimality of the initial plan (x, T ). Therefore, the appropriate isopropfit line must be tangent to V 1 at (x 1,T 1 ), which implies that the smaller package must also be efficient. Both the statement with inequality (T 2 T 1 ) c (x 2 x 1 )andthecasewithequality (T 2 T 1 )=c (x 2 x 1 )canbeproveninsimilarway(inthelattercase,onecanalsouse the fact, proven by contradiction, that at least one of the two packages must be efficient). 8
10 Q.E.D. Note that the overall socialefficiency can be assured when both net tariffs are the same. The equality of gross tariffs T 2 = T 1 by itself is not sufficient for overall efficiency. Proposition 2 can be generalized for the case of several consumers. Indeed, take n consumers satisfying inequalities like (T i+1 T i ) c (x i+1 x i )fori n. Ifthelargest package contains efficient quantity, then one can argue, as in Proposition 2, that the secondlargest package also contains efficient quantity. Further, using induction we can conclude that all packages contain efficient quantities. Similarly arguments can be used for inequality (T i+1 T i ) c (x i+1 x i ). When all nettariffs are the same it is not necessary to know whether any package is efficient. In other cases, our proposition enables to only link one efficient package with another. When demands are ordered, the well known result in the literature is that the largest package must contain an efficient output. Although theoretically there may be other possibilities under nonordered situations, in general, the assumption that the largest package is efficient is not unrealistic. 4. Conclusions For nonordered valuations, two different incentivecompatible and profit maximizing packages offered to two types of consumers could simultaneously be socially efficient and firstbest. It shows that a monopolistic nonlinear price discrimination practiced by offering different sized packages is compatible with overall allocative efficiency. Therefore, onecanarguethatthereareprobablesituations when a legal case against a monopolist 9
11 using seconddegree price discrimination may not be strong enough, at least on efficiency grounds. Under reasonable assumptions, when making inferences about sociallyefficient discriminations, it is sufficient to observe package sizes, tariffs and costs. References Rochet, J.C. and L. Stole (2003): The Economics of Multidimensional Screening, in Advances in Economics and Econometrics: Theory and Applications, Eighth World Congress, Wilson, R.B. (1993). Nonlinear Pricing. Oxford: Oxford University Press. 10
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