Two-Part Pricing, Public Discriminating Monopoly and Redistribution: A Note

This note analyzes some properties of optional two-part pricing in a two-type economy. First, the optimal contracts along the Paretian frontier are described. Then, the duality relation between the Rawlsian program and the discriminating monopoly is demonstrated. Last, this property is used to build a mutualist mechanism implementing the constrained Pareto optima.


INTRODUCTION
Optional two-part pricing, extensively used by many public utilities (electricity, water, railways etc.), gives, as shown by Sharkey and Sibley (1993), some freedom to redistribute the social surplus. 1 These authors show in a partial equilibrium framework that, when a monopoly proposes a menu of contracts, each specifying the fee and the charge price, it is possible for a social planner controlling this monopoly to redistribute towards the weak demand consumer. Our contribution is not to extend their study to a new framework or to other pricings. Our ambition is, ®rst, to emphasize the redistributive mechanism of optional two-part pricing, notably with the help of some graphical presentations, and, second, to propose a simple incentive mechanism implementing the more redistributive optima. As we shall see, this mechanism takes advantage of the dual relationship between the program of the discriminating monopoly and the social planner's program.
This note is organized as follows. The economy is described in section 2. The constrained Pareto optima are characterized in section 3 even though an implementing mechanism is proposed and studied in section 4. Limits and possible extensions of this work are discussed in the last section.

THE ECONOMY
There are two goods, the produced good and the nume Âraire good. Their quantities are respectively denoted q and w. The economy is composed of two types of agents, indexed i 1, 2, de®ned by their quasi-linear utility functions: The functions V i verify the following properties.
Assumption 1: V i is continuously twice differentiable with and V 9 2 (q) . V 9 1 (q) Relations (1) state that the inverse demand is positive and strictly decreasing. Relation (2) implies that type 2's demand is higher than type 1's; for quasi-linear utility functions this relation is also the standard single crossing assumption.
The cost function C of the monopoly which produces the good q veri®es the following assumption.
Assumption 2: C is a convex function on ]0, I[: 2 The monopoly using this technology proposes two contracts ( p 1 , E 1 ) and ( p 2 , E 2 ), where p 1 , p 2 are the usage charges, E 1 and E 2 the fees. As asymmetric information prevents perfect discrimination, the contracts must be incentive-compatible. Moreover, in order to eliminate trivial cases, we will suppose that ®rst-best optima are characterized by strictly positive consumptions.

CONSTRAINED PARETO OPTIMA
In this section, the constraints regimes of the Paretian program are speci®ed. Then, the Pareto frontier is outlined and the main properties of optimal contracts are discussed. This section does not propose any new results; its aim is simply to clarify the characterization of the Pareto frontier and above all to present, with the help of a graph, a pedagogical analysis of the redistributive mechanism of optional two-part pricing.
In our economy, the constrained Pareto program P p (s 2 ) is 2 ) n 1 D 1 ( p 1 ) n 2 D 2 ( p 2 ) and n i is the number of agents of type i.
To discuss the constraints regimes of P p (s 2 ), it is useful to consider the ®rst-best optima which verify the incentive constraints. Actually, as for every ®rst-best optimum, prices are equal to the marginal cost, and incentive constraints require equality of fees. Hence, there is a unique ®rst-best optimum which veri®es the incentive constraints, the so-called Coase two-part pricing. The types surpluses at the Coase solution are noted s co 1 and s co 2 . 3 In the following, only the domain s 2 , s co 2 is studied. 4 In this domain the binding incentive constraint is IC 2 . 5 To know if SC 2 binds, it is useful to introduce the Rawlsian solution de®ned by the maximization of the type 1 surplus subject to incentive and budget constraints. As these constraints always bind, the Rawlsian objective function, after some substitutions, can be rewritten where the social surplus S s ( p 1 , ]X As usual in this literature, we assume the concavity of this function, and hence the unicity of the Rawlsian solution ( p R 1 , p R 2 ). If s R i is the surplus of type i at this Rawlsian optimum, 6 two cases must be distinguished depending on whether s 2 is above or below s R 2 . For s R 2 < s 2 , s co 2 , 7 the constrained Pareto program is equivalent to maximizing W R ( p 1 , p 2 ) subject to SC 2 . The (assumed) strict concavity of W R implies two results. First, SC 2 is binding; second, the second-best frontier, in the surplus space (s 1 , s 2 ), is continuous and strictly monotonic (see ®gure 1).
The remaining question is what the properties of the optimal contracts are along the second-best frontier. As only incentive and participation constraints of type 2 bind for s R 2 < s 2 , s co 2 , the program P p (s 2 ) is reduced to the following one: max p 1 , p 2 1 n 1 n 2 fS s ( p 1 , p 2 ) À n 2 [S 2 ( p 1 ) À S 1 ( p 1 )]g subject to The other case is symmetrical. To extend our results to the domain s 2 . s co 2 , we only need to rewrite program P p (s 2 ) by permuting indices, i.e. we need to maximize s 2 subject to the participation constraint of type 1. 5 The space being limited here and the proof being classical, it is not reproduced in this note. 6 As by assumption s co 1 is strictly positive, s R 1 . 0. Then, from IC 2 and equation (2), it can be shown that s R 2 . s R 1 . 0. 7 Of course, for s 2 , s R 2 , as s R 2 is the lower value that the Paretian social planner can actually assign to type 2, SC 2 of program P p (s 2 ) is always released.
For each s 2 , the ®rst-order conditions give optimal prices: where Rm 1 (q 1 ) V 9 1 (q 1 ) q 1 V 0 1 (q 1 ) is the marginal revenue upon type 1 and ë is the Lagrangian multiplier. 8 Equation (5) re¯ects the fact that p 2 is not an incentive tool when one tries to increase the type 1 surplus above this Coasian level. 9 Second, we S 2

Second-best frontier
Coase solution

Rawlsian solution
First-best frontier 45º S 1 Figure 1. The frontier of the constrained Pareto optima with optional two-part pricing.
8 Equation (6) can be rewritten as follows: We note that for ë n 2 an 1 we obtain the Coasian prices: Otherwise, one could show that for ë 0, p 2 p R 2 . So, intuitively, we could interpret ë as a relative weight of type 2 in a linear social welfare function; but this interpretation assumes the concavity of the Pareto frontier (in the surplus space). 9 This is a well-known result of adverse selection models with the Spence±Mirrlees assumption (relation (2) of assumption 1). In a similar framework with n types of agents, Sharkey and Sibley (1993) prove the same result.
In ®gure 2, the Coasian equilibrium is depicted by points A and B in the space (q, T ) where T is the total spending of each type. The upward line passing through these points is the nil pro®t line; it is also the spending line of each type when the fee is E and the price is equal to the marginal cost c. 10 The curves passing through A and B are the isosurplus curves corresponding respectively to types 1 and 2.
At the Coase optimum, and in fact at each constrained Pareto optimum, the only way to increase s 1 is obviously to decrease s 2 . Nevertheless, for this, one needs to release the incentive constraint of type 2, i.e. to decrease the surplus of the dishonest type 2. Starting from the Coasian point A, the only way to proceed is to raise p 1 (with an appropriate adjustment of E 1 leaving s 1 constant). 11 As the surplus of the dishonest type 2, reached at point F, is now only s9 2 (, s 2 ), the social planner can extract at most dð 2 with the new contract (E9, c). As we can see in ®gure 2, the increase of budget surplus dð 2 over type 2 exceeds the budget loss dð 1 over type 1. 12 So starting from the Coase equilibrium, such an adjustment leaves a positive net budget surplus which, equally redistributed to check incentive constraints, increases type 1's surplus. 13 Since it permits more redistributive surplus allocation than the Coase solution, optional two-part pricing is a useful tool for the social planner. But, if he does not directly control the monopoly, implementation of the 10 For simplicity, we supposed in this graph that the marginal cost is constant. 11 Indeed, it is easy to see that a decrease of p 1 (leaving s 1 constant) incites type 2 to lie, increases s 2 , and breaks the budget constraint. 12 In fact, to ®rst order, dð 1 is negligible, which is not the case for dð 2 . 13 Those adjustments can be reproduced for all constrained Pareto optima but the Rawlsian one. constrained optimum is questionable: how can he induce the monopoly to select the right two-part pricing?

IMPLEMENTATION BY DISCRIMINATING MONOPOLY
In this section we build mechanisms which implement the more redistributive optima. To reach this aim we study a regulated monopoly, the so-called monopoly a Â la Edgeworth. 14 This monopoly is supposed to use optional two-part pricing and is subject to an additional constraint to leave a minimal surplus to type 1. The implementing mechanisms are then deduced from the duality relation between the discriminating program of this monopoly and the Rawlsian program; this duality relation was incidentally noticed by Roberts (1979, pp. 80±81) in a continuous types economy for nonlinear pricing. Before introducing the monopoly a Â la Edgeworth, let us ®rst introduce the program P m of the simple discriminating monopoly using optional two-part pricing and begin to show that the Rawlsian prices are also the monopolistic ones.
Proposition 1: The Rawlsian prices ( p R 1 , p R 2 ) are the solutions of the monopoly program.
Proof: Under assumption 1, IC 2 and PC 1 are the only active constraints and we get Hence, after substitutions, the objective function becomes The end of the proof is now obvious.
j Consequently, the monopoly equilibrium and the Rawlsian solution differ only by E 1 and E 2 . In fact, this result hides a fundamental link between them: they are the two polar solutions of the monopoly a Â la Edgeworth. The latter is a discriminating monopoly with an additional surplus constraint for type 1: As one can easily demonstrate using classical arguments, equation (2) of assumption 1 implies that SC 1 and IC 2 are the only binding constraints. After manipulations, the previous program is reduced to the following free maximization: max p 1 , p 2 (n 1 n 2 )[W R ( p 1 , p 2 ) À s 1 ] Optimal prices and quantities are independent of the s 1 level and equal to the monopoly ones. By raising s 1 (from 0 to s R 1 ), all surplus distributions between the monopoly equilibrium and the Rawlsian solution can be achieved. Actually, for s 1 s R 1 , the program of the monopoly a Â la Edgeworth is the dual of the Rawlsian program. So, naturally, it gives not only the same prices but also the same fees. Of course, the monopoly a Â la Edgeworth is an abstract mechanism since s 1 is exogenous.
A way to make the mechanism more realistic is to consider a mutualist mechanism, i.e. a pro®t-sharing device. In our framework, one can view a mutualist monopoly as a ®rm which redistributes all its pro®t to its members 15 according to a sharing key. If this key is contingent upon the chosen contracts, membership guarantees a part of the pro®t even if the member does not consume. Furthermore, we will suppose that this sharing key is ®xed ex ante and the pro®ts are redistributed ex post. The mutualist monopoly's customers are thus considered just as shareholders. Therefore, it is natural to suppose that the aim of the mutualist monopoly is to maximize pro®t. Finally, for a sharing key (è 1 , è 2 , è) the program P mu (è 1 , è 2 , " è) of this monopoly is then subject to PC 1 : where èÐ is the guaranteed share pro®t and è i Ð is the pro®t share of type i.
Proposition 2: If the monopoly pro®t is uniformly distributed, è 1 è 2 è, the mutualist monopoly equilibrium gives the Rawlsian surplus to each type.
Proof: With the uniform sharing key, the program is reduced to the program P m . So the optimal quantities and fees are the Rawlsian ones. j The intuition of the previous proposition can easily be grasped graphically (see ®gure 3). Points A and B correspond to the private monopoly equilibrium (where s 1 0). 16 Because of the quasi-linearity of preferences, a uniform monetary transfer (to both types) implies a vertical translation of the Edgeworthian monopoly equilibrium: when a surplus s 1 is granted to type 1, the private equilibirum is translated to the new equilibrium represented by A9 and B9.
Furthermore, the previous mechanism suggests its extension to the set of constrained Pareto optima with s R 2 , s 2 , s co 2 .
Proposition 3: For every s 2 , with s R 2 , s 2 < s co 2 , there exists a price cap p such that the mutualist discriminating monopoly mechanism implements quantities and prices of the constrained Pareto optimum corresponding to s 2 .
Proof: With the price cap p, the monopoly program becomes Assumption 1 always implies that IC 1 and IC 2 cannot both be binding. As PC 2 is released, IC 2 must bind and IC 1 is then loosened. So, PC 1 is active and the previous program is reduced to max p 1 , p 2 S s ( p 1 , p 2 ) À n 2 [S 2 ( p 1 ) À S 1 ( p 1 )] subject to PCC i : For right values of p ( p co 1 < p < p R 1 ), this program implies, for every value of p 1 , p 2 C9. And by strict concavity, p 1 p. To implement constrained Pareto optima (in quantities and prices) for s R 2 , s 2 < s co 2 , it is suf®cient to set p p p 1 (s 2 ), where p p 1 (s 2 ) is the optimal price p 1 of program P p (s 2 ). 17 j

CONCLUSION
This note explores the redistributive properties of optional two-part pricing in a two-type economy. It shows that a monopolistic structure market augmented by a uniform pro®t sharing allows one to implement the most redistributive optimum, the Rawlsian solution. If a price cap is added, this mutualist mechanism allows one to achieve less redistributive constrained optima. However, there are three caveats to bear in mind.
First, in a pure mutualist mechanism, only customers share pro®t, even though, in the proposed mechanism, each agent receives pro®t independently of his consumption decision. However, as here each agent is a customer, the difference is blurred. So, this mechanism can only be applied to a subset of quasi-universally consumed goods, such as electricity, water, public transport.
Second, the ef®ciency of this mechanism requires of course the social planner to have such precise information as to prevent managers and employees from capturing pro®ts. So, the mechanism supposes a strict monitoring of the managers.
Last, a strong implicit assumption of this paper is the fact that the social planner has a unique redistribution tool: public pricing. Of course, in a more general framework he can also use income taxation. So, a natural extension would be to study the complementarity between discriminating public pricing and income taxation. 18