Pricing discretion and price regulation in competitive industries
∗Alberto Iozzi†
Universit`a di Roma
“Tor Vergata”
Roberta Sestini
Universit`a di Roma
“La Sapienza”
Edilio Valentini
Universit`a “G. D’Annunzio”
di Chieti
October 24, 2002
Abstract
This paper analyses the effects on prices, entry and social welfare of different regulatory regimes limiting the pricing discretion of a reg- ulated firm facing (potential or actual) competition only in a subset of markets it supplies. We focus on two rules, both including a price cap on the average level of prices set by the regulated firm. One of the two regimes (Absolute) places a fixed upper limit to the price charged in the monopolist market; the other regime (Relative) entails a limit on the captive price relatively to the price set in the competitive mar- ket. We find that the Relative regime induces a (weakly) lower price in the captive market but a (weakly) higher price in the competitive market, so that entry is more likely to occur under this regime. The welfare ranking between the two regimes is ambiguous and depend on the welfare measures adopted, on the degree of competitiveness of the competitive market, as well as on the values of the parameters. In general, the more competitive is the market in case of entry the more likely is that the Relative regime is socially preferred.
JEL Numbers: L13, L50.
Keywords: price regulation, price discrimination, entry.
∗The authors wish to thank Nicola Rossi and Carla Pace for their contributions in the early stages of this project, and Gianni De Fraja, Massimo Roma, Carlo Scarpa, and participants to University of Rome Tor Vergata 2001 Summer School, NIE 2001 Annual Meeting, V Congress on the Economics of Telecommunications, 2001 EARIE Conference, 2002 Workshop on ’Comportements opportunistes et dysfonctionnement ´economique’ in Li`ege’ and to a seminar at the University of Padua for their useful comments
†Corresponding author: Universit`a di Roma “Tor Vergata”, Dip. SEFEMEQ, Via Columbia 2, I-00133 Rome, Italy. E-mail: Alberto.Iozzi@UniRoma2.It
1 Introduction
A very important feature of regulatory policies is how they take into ac- count the fact that the regulated firm typically operates in markets where the degree of competition - actual or potential - may dramatically differ.
Indeed, this is a typical characteristic of most regulated industries, where the development of competition has occurred at a different pace in the dif- ferent markets and where some markets are still captive for the regulated dominant firm.
While price cap regulation is by now the rather standard instrument to limit the overall level of the prices set by the regulated dominant firm, different approaches have been taken by regulatory authorities to face the many problems deriving from the different competitiveness of the markets where the regulated firm operates. An important problem that regulators have to tackle when devising their regulatory policies regards the fact that the regulated firm may exploit the very large degree of pricing discretion it enjoys under price cap regulation with possible anti-competitive conse- quences (Armstrong and Vickers, 1993; Otero and Waddams, 2002). As consequence, the choice that regulators have to make regards the nature of the limits (if any) they want to impose on the discretionality of the regulated firm over its price structure.
The traditional solution worked out in the practice of regulation is to im- pose additional caps over the prices in the captive markets in order to limit the regulated firm’s ability to obtain extra revenues. Another approach to limit the pricing discretion of the regulated firm entails cross-market re- strictions on prices charged in markets with different competitive features.
For instance, ex-Public Electricity Suppliers in the UK (the former regional monopolists in the distribution activity) are subject to an average price cap and also face relative price regulation, which links the prices charged for prepayment customers (thought to be not sufficiently protected by compe- tition) to those charged to credit customers (i.e. those customers acting in markets where competition is sufficiently developed).
This paper compares these two alternative approaches both in terms of fostering competition and maximising social welfare, which are usually the primary objectives of regulators. More precisely, we focus on the effects on entry and welfare of two different regulatory regimes that, together with an average price cap, either set an fixed upper limit on the captive price chosen
by the regulated firm or place a limit to this price that depends on the price set in the competitive market.
The relevance of the issue of the limits to the pricing discretion by a reg- ulated firm has been recently emphasised by Armstrong and Vickers (2000) and Armstrong and Sappington (forthcoming). The first paper recognises different possible motivations for limiting the pricing discretion by a regu- lated firm, but mainly focus on the effects of these limits on allocative effi- ciency in a monopoly. Other papers on the same issue are Armstrong and Vickers (1991), who investigate the welfare effects of price discrimination by a monopolist subject to different average price constraints, and Ireland (1992), who analyse how relative price regulation performs in those situa- tions where asymmetric information prevents regulation upon the absolute level of prices.
All these studies however do not consider the possible implications upon competition of the limits to the pricing discretion of a dominant firm. To our knowledge, this issue is first tackled in Armstrong and Vickers (1993) who analyse, in a two-market setting, the effects of different policies toward price discrimination by a dominant firm that faces potential competition in one of the markets it serves, also in the presence of price cap regulation. They show that entry is weakly more likely when the pricing discretion by the dominant firm is constrained by a ban on price discrimination and when the firm faces independent caps in the two markets; both these policies have however ambiguous effects on welfare. More recently, Anton et al. (2002) focus on the issue of cross-market restrictions associated with universal services provisions. In a two-market setting - one captive, one where two firms interact strategically -, they find that these cross-market restrictions are at a disadvantage of the firm supplying both markets. Other papers more closely concerned with the effects on competition of price cap regulation are those of B¨os and Nett (1990) and Iozzi (2001). Bos and Nett study a duopoly game with quantity pre-commitment where one of the firms is price capped. They find that the lower the cap the lower is the capacity and output of the entrant. Iozzi (2001) studies the issue of entry deterrence under dynamic price cap regulation; he shows that this form of regulation allows the regulated firm to deter entry by acting as a commitment device for a more aggressive pricing policy in case of entry.
Our analysis is then complementary to Armstrong and Vickers (2000)
in that we analyse the effects that these constrains on pricing discretion may have on the development of competition and social welfare rather than on allocative efficiency in a monopolistic market. Moreover, it follows the approach adopted by Armstrong and Vickers (1993) but differs in that it examines how competition and welfare are affected by the adoption of dif- ferent policies aimed at limiting the pricing discretion by the regulated firm rather than by the adoption of different forms of price caps.
We focus on a simple set-up where a regulated firm serves two markets with identical demand functions, differing only for the viability of competi- tion. In the potentially competitive market, a price-taker firm may enter by paying a fixed entry cost and choosing an exogenously given scale of entry. A benevolent regulatory authority may choose between two regulatory regimes that constrain only the pricing choices of the incumbent firm. Both regimes include a price cap on the average level of its prices. Moreover, one of the two regimes - referred to as the Absolute regime - places an fixed upper limit to the price charged in the monopolist market; the other regime - referred to as the Relative regime - places instead a limit on the captive price relatively to the price charged in the competitive market, i.e. it constrains the ratio of the two prices.
We study the effects of these two regulatory regimes upon equilibrium prices, the entry decision and social welfare. The main results of the paper are as follows. The Relative regime induces the regulated firm to charge a (weakly) lower price in the captive market but, on the other hand, causes a (weakly) higher price in the competitive market. Hence, the Relative regime, by constraining the regulated firm to lower the price in the captive market when reducing the price in competitive market, transfers to the monopo- listic market some of the downward pressure on prices which operates in the competitive market. However, this is done at the cost of keeping the competitive price higher than it would be in the Absolute regime. This less aggressive pricing response in the competitive market under the Relative regime makes entry more likely to occur; indeed, potential profits for the entrant are higher as well and entry occurs for a wider range of entry costs.
Making use of specific functional forms for demand and cost, we also assess the welfare properties of the two regimes. In general, the ranking between the two regimes is ambiguous and depends on the welfare measures adopted and on the degree of competitiveness of the competitive market -
as approximated by the scale of entry of the potential entrant and by its elasticity of supply -, as well as on the values of the parameters.
More specifically, when entry occurs only under the Relative regime, this regime makes consumers better off and firms worse off. Social welfare tends to be higher under the Absolute regime, unless the degree of competition faced by the regulated firm under the Relative regime places on its prices a strong downward pressure, so that the positive effect of this regime on consumers outplays its negative effect on profits. On the other hand, when entry occurs under both regimes, consumers’ surplus tends to be higher under the Relative regimes and firms’ profits tends to be higher under the Absolute regime, while the effect on aggregate welfare in indeterminate.
The instances in which the Relative regime is preferred tend to increase the larger is the scale of entry, the slacker is the constraint on relative prices, the smaller is unit cost of the incumbent firm and, more importantly, the more elastic is the supply of the entrant.
The paper is organised as follows. Section 2 illustrates the model and describes the features of the two regulatory regimes under analysis. Section 3 provides some general results on equilibrium prices and entry under the two regimes; an illustration of these results is also given using an example with linear demand and price constraints. Section 4 employs two linear examples to investigates the welfare properties of two schemes. Some final remarks are given in Section 5. All the proofs are relegated to the Appendix.
2 The model
We employ a very simple two-market model based on Armstrong and Vickers (1993). Price in each market is denoted by pi (i = 1, 2). Demand is given by x(pi) and is assumed to be independent and symmetric across markets, that is demand in each market does not depend on the price set in the other market and both markets have the same demand function. While independence is assumed for the sake of simplicity, we impose symmetry in order to leave out any differences across markets that are not due to the regulatory rules and the different likelihood of entry. Furthermore, x0(.) < 0 and x(p) = 0 for any p ≥ pmax, where pmax∈ <++.
Consumers have quasi-linear utility functions, therefore demands are in- dependent of income. Roy’s identity implies that x(pi) = −v0(pi), where
v(pi) denotes aggregate consumers’ surplus in each market.
An incumbent profit maximising monopolist, firm M, operates in both markets. Firm M has constant unit cost c in each market. We denote firm M’s profits in market i by πi(pi) = x(pi)(pi− c) for i = 1, 2. A potential new entrant, firm E, may enter market 2, but not market 1 where sunk costs are so high that entry is not profitable. If firm E enters, it operates as a price taker to maximise profits. We denote output with ks(p2), where k is the scale of entry and s(p2) is the supply function per unit of capital.
We assume that the value of k is chosen by the entrant amongst zero and an exogenously given value K, so that k ∈ {0, K} where K ∈ <++. By Hotelling’s lemma we obtain that s(p2) = e0(p2) where e(p2) gives firm E’s profit per unit of capital, with e0(.) > 0 and e00(.) ≥ 0 . We denote by f (k) the cost of entry at scale k and assume that f (0) = 0 and that, for any K1
and K2 such that 0 < K1 < K2, f (K1) < f (K2).
We hypothesise that there exists a benevolent industry regulator who chooses between two regulatory regimes that, as it is common in practice, only constrain the prices set by the dominant firm (in this paper, firm M) but not those of the (potential) new entrant. Both regimes feature a price cap constraint; we simply assume that this constraint takes the form pi ≤ ϕ(pj), where ϕ(.) is a strictly decreasing and symmetric function, for i, j = 1, 2 and i 6= j; this is such that it would never allow firm M to set unconstrained profit maximising prices. Each of the two regulatory regimes also includes an additional constraint which, by placing an upper limit to the captive price, limits the pricing discretion of the regulated firm to prevent it to exploit its monopolistic condition in this market. In one case, this additional constraint takes the form p1≤ p1; quite naturally, we assume that p1≤ pm1 , where pm1 is the unconstrained profit maximising price in the captive market. We define the regulatory regime given by the combination of this constraint with the price cap as the Absolute regime. Alternatively, the additional constraint can take the form p1≤ b(p2), where b(.) is a strictly increasing function. This regime is referred to as the Relative regime. In other words, the Absolute regime is such that firm M is constrained to choose the prices in the set PA ≡ {(p1, p2) : p1 ≤ ϕ(p2) and p1 ≤ p1}, while the set of allowed prices under the Relative regime is given by PR ≡ {(p1, p2) : p1 ≤ ϕ(p2) and p1 ≤ b(p2)}. We assume that the constraints under both regimes are such that
equilibrium prices exist and are always unique.1 Furthermore, to ensure an easier comparability of the two regimes and to formalise the idea that the regulator’s view over the maximum allowed price in the monopolistic market is independent from the choice of the regime, we assume that the maximum allowed level for p1 is identical under both schemes.2
Figure 2 illustrates in the price space the sets of admissible prices under the Absolute and the Relative regime respectively for the special case of linear constraints. The price cap which is part of both regimes is depicted by the downward sloping line AB which features in both panels of the Figure.
Panel A illustrates how the Absolute regime constrains the firm to choose prices within the shaded region below the average price cap line AB and the horizontal line CD which intercepts the y-axis at p1. Similarly, the shaded area in Panel B under the price cap line AB and the upward sloping line OE illustrates the combination of prices allowed under the Relative regime. The dotted line linking the two panels illustrates the effects of assuming that the two additional constraints limiting the pricing discretion of firm M allow a highest p1 identical under the two regimes.
We deal with a finite game of perfect information with the following order of moves:
• stage 1: the regulator chooses either the Absolute or the Relative regulatory regime;
• stage 2: firm E chooses the scale of entry k ∈ {0, K};
• stage 3: firm M chooses p1, p2 ∈ [0, pmax] subject to the regulatory regime selected by the regulator;
• stage 4: firm E chooses the optimal quantity per unit of capital.
Hence, first the regulator chooses the regulatory regime. Then, the potential entrant chooses whether or not to enter by paying the sunk cost of entry. In the subsequent stage, the incumbent chooses its optimal prices subject to the regulatory regime in place. Finally, if entry has occurred, the entrant
1Intuitively, this requires that b(.) and ϕ(.) are not ’too’ convex relatively to the shape of the iso-profit contours.
2Formally, this implies that p1 = p01, where p01 is the solution with respect to p1 of p1 = b(ϕ(p1)). This, together with the fact that b(.) is strictly increasing, implies that b(0) ≤ p1.
Figure 1: Prices allowed under the two regulatory regimes
selects its optimal quantity. Given the nature of the game, we solve it by backward induction.
For any scale of entry k ∈ {0, K} chosen by firm E, the incumbent’s profits are given by
π(p1, p2) = π1(p1) + π2(p2) − ks(p2)(p2− c), (1) while profits of firm E are given by
θ(p2) = ke(p2) − f (k). (2)
The regulator’s payoff is given by the social welfare function
W (p1, p2) = v(p1) + v(p2) + α(π(p1, p2) + θ(p2)) (3) where α ∈ [0, 1] expresses the distributional concern of the regulator.
3 Effects on equilibrium prices and entry of the different regulatory regimes
This section derives some properties of the equilibria of the two market subgames between the firms under the two regulatory regimes. We start from studying some properties of the equilibrium prices under the different regimes; then, using these results, we investigate the effects of the different regulatory regimes on the entry choice of the potential entrant. Consider now the problem faced by the incumbent. This is given by
maxp1,p2
π1(p1) + π2(p2) − ks(p2)(p2− c) (4) s. t. p1, p2∈ Ph
where h = A, R. Notice that, in making its choices, firm M already takes into account the optimal response of the rival in the last stage of the game, as given by its net supply function. Also, this is the problem faced by the incumbent both with and without entry. It is indeed sufficient to set k equal to zero to have the problem faced by the regulated firm when operating as a monopolist.
Let ph1 and ph2 (where h = A, R) be the prices that solve problem (4).
Then,
Proposition 1. pA2 and pR2 are decreasing in K.
This result, which holds also in more general settings3, depends on the fact that the elasticity of the residual demand faced by firm M increases as K increases. Being the optimal price inversely related to the elasticity, the larger is the scale of entry the lower is the price charged by the regulated firm in the competitive market. We can also state
Proposition 2. pA1 ≥ pR1 and pA2 ≤ pR2. Moreover,
Corollary 1. ∃ eK ≥ 0 such that, for K ≤ eK, pAi = pRi , for i = 1, 2.
While Corollary 1 states that there exists a threshold value for the scale of entry K of the potential entrant firm below which the choice of the regula- tory regimes does not affect the equilibrium of the market game, Proposition
3This result has been indeed derived by Armstrong and Vickers (1993) in a similar set-up.
2 illustrates that, provided that K is above this threshold level, the equi- librium price in the competitive market is higher under the Relative regime than under the Absolute regime. Under the same conditions, the equilib- rium price in the captive market under the Absolute regime is weakly higher than under the alternative regime. Intuitively, when K is sufficiently large, any price reduction under the Relative regime induced by entry in the com- petitive market comes to firm M at the additional cost of a price reduction also in the captive market. This leads firm M to respond to entry in a less aggressive manner than it would do under the Absolute regime. On the other hand, the lower captive price under the Relative regime is due to the ability of this regime to transfer to the captive market the price reductions which take place in the other market. Clearly this is not the case under the Absolute regime, where the captive price is determined independently from the competitive price.
To better illustrate these results we provide the following example.
Example 1: Linear demand and inelastic supply function
In this example, demand in both markets is given by x(pi) = a − pi, for i = 1, 2. We assume that firm E’s supply function per unit of capital is constant and always equal to 1, e. g. s(p2) = 1. Using Hotelling’s lemma, the profit function per unit of capital is given by e(p2) = p2. We hypothesise that all the regulatory constraints are linear. In particular, the price cap constraint under both regimes takes the form p1+ p2 ≤ a.4 Notice that this constraint grants that the regulated firm is never able to set unconstrained monopoly prices when c > 0. The constraint on relative prices under the Relative regime is assumed to take the form p1 ≤ βp2. The assumption made in the general set-up that the maximum allowed level for the monopoly price is identical under both schemes, translates here into the possibility to write p1
in terms of β, e. g. p1= 1+βaβ .
We constrain the choice of p1 so that p1 ∈ [a2,a+cc ]; notice that an equiv- alent restriction is placed on the admissible range of β. This assumption grants that, on the one hand, the maximum level of the price in the cap- tive market is at most equal to the unconstrained monopoly price, and, on the other hand, that the regulatory regimes never force the firm to set a
4This derives from assuming a price cap constraint given by wp1+ (1 − w)p2≤ p, where we assume p =12a and, to impose symmetry, w = 12.
monopoly price lower than the price in the competitive market. Finally, in order to rule out some pathological parametric conditions we assume that c < 14a; this ensures that the regulated firm always makes nonnegative profits and that it always faces a non-negative residual demand curve at equilibrium prices.5
The optimal prices chosen by firm M in stage 3 under the Absolute regime are as follows:6
pA1 = a2 +K4, pA2 = a2 − K4 for 0 < K < K0; (5) pA1 = β+1aβ , pA2 = β+1a for K0 < K < K00; (6) pA1 = β+1aβ , pA2 = a+c−K2 for K00 < K < KmaxA ; (7) where K0 = 2a(β−1)β+1 , K00 = a + c − β+12a and KmaxA = a − c, where KmaxA is the highest scale of entry compatible with equilibrium non-negative residual demand for firm M.
Intuitively, when the regulated firm is a monopolist in both markets, it chooses the prices along the 45◦ degree line which are just allowed by the average price cap constraint. If entry has instead occurred at a sufficiently small scale, the regulated firm finds it profitable to charge a price in the competitive (respectively, captive) market lower (respectively, higher) than the price it would choose without entry. Since the (linear) average price cap is binding, both prices linearly depend on K (with opposite sign), and the wedge between the two prices increases the larger is the scale of entry of the competitor. When the scale of entry is sufficiently large, the price that would be charged in the captive market is not permitted by the constraint on the absolute level of this price, which is then always binding irrespective of the level of the price in the competitive market. For sufficiently large values of K, the latter price decreases as K increases, while the former is independent of K. As a matter of fact, when the scale of entry is large enough, the average price cap is not binding and the pricing choices of the regulated firm are limited only by the constraint on the absolute level of the captive price.
5Notice that this condition on c is more restrictive than necessary for the existence of the equilibrium under the Relative regime, for which it would suffice that c < 13a.
6Details are available from the authors upon request. Notice these are also the equilib- rium prices in the absence of entry, since they can be obtained simply by setting K equal to zero in these expressions.
Figure 2: Equilibrium prices in Example 1 (c = 0.2, β = 1.05 and a = 1)
Similarly, under the Relative regime we find that
pR1 = a2 +K4, pR2 = a2 −K4 for 0 < K < K0; (8) pR1 = β+1aβ , pR2 = β+1a for K0 < K <; K000 (9) pR1 = β[(a+c)(β+1)−K]
2(β2+1) , pR2 = (a+c)(β+1)−K
2(β2+1) for K000 < K < KmaxR ; (10) where K0 takes on the same value as before, K000 = (a + c)(β + 1) −2a(ββ+12+1) and KmaxR = 2aβ2+a−c−β(a+c)
1+2β2 . Notice also that KmaxR < KmaxA .
When entry has not occurred or has occurred at a sufficiently small scale, optimal prices are identical to those set under the Absolute regime; in other words, in this example we have that eK = K00. On the other hand, when K is sufficiently large, the price set by the regulated firm in the competitive market decreases with the scale of entry. Clearly, this is also what occurs under the Absolute regime. However, under the Relative regime, because of the additional constraint on the price ratio, if the regulated firm charges a low price in the competitive market it is constrained to charge a low price in the captive market as well. Hence, the larger the scale of entry, the lower
is the price in the competitive market but also, differently than under the Absolute regime, the lower is also the price in the captive market.
Equilibrium prices under the two alternative regimes are plotted in Fig- ure 3 against the whole admissible range of K values. For intermediate values of K (i.e. K00 ≤ K ≤ K000), while the level of the monopoly price is identical under both regimes, the competitive price is higher under the Relative regime. When K is sufficiently large (i.e. K000 ≤ K ≤ KmaxR ) the higher competitive price under the latter regime is coupled with a lower monopoly price.
We now go back to the general case, turning to the analysis of stage 2, when the optimal choice of entry is taken by firm E. Firm E chooses to enter and pays the entry cost f (K) only if it anticipates non-negative profits in the subsequent stages of the game, that is only if θ(ph2, K) = Ke(ph2) − f (K) ≥ 0,where i = A, R. The effect of the regulatory regime on this choice is illustrated in the following Proposition, which follows immediately from Proposition 1.
Proposition 3. Let φA(K) and φR(K) be the highest level of entry cost which makes entry profitable under the Absolute and Relative regime respec- tively. Then, φA(K) ≤ φR(K).
This Proposition states that, provided that the scale of entry is large enough, the Relative regulatory regime is able to foster entry for a larger range of values of the potential entrant’s entry cost. The key reason for this result is that the equilibrium price in the competitive market is always higher under the Relative regime than under the Absolute regime. Hence, the entrant’s profits in case of entry are higher under the Relative regime.
This may lead the potential entrant to choose to enter under cost conditions that would make entry unprofitable under the alternative regime. However, since this higher likelihood of entry is obtained by trading off higher prices in the competitive market with (weakly) lower prices in the captive market, the overall effect of the two regulatory regimes is not clear-cut. The analysis of the welfare consequences of the two regulatory regimes is carried out in the next section.
4 Welfare analysis
In this section we analyse the choice of the regulator at stage 1 of the game;
we focus on cases when K is within a range such that the choice by the regulator has differentiated effects on the equilibrium of the game and, at the same time, an equilibrium exists under both regimes.
The welfare consequences of the two regulatory rules are the result of different effects, both on the firms’ profits and on consumers’ surplus. In terms of profits, the Relative regime undeniably benefits the new entrant firm since the response to entry is less aggressive but, at the same time, it may be to the detriment of the incumbent. In terms of consumers’ surplus, the Relative regime causes a higher competitive price than the Absolute regime but a lower price in the captive market. This complex interplay prevents us from obtaining results as general as in the previous section. We then provide findings obtained by making use of linear examples. In some cases, the analysis of the welfare properties of the two regimes is carried out by means of numerical simulations.7 We argue that this methodology does not unduly restrict the significance of our analysis. As a matter of fact, we aim at finding plausible regularities in the conditions that ensure that one regime performs better than the other in terms of guaranteeing higher social welfare. Numerical analysis is, in our view, an appropriate, albeit maybe inelegant, instrument for this purpose.
Example 1: Linear demand and inelastic supply function
Everything is as in Example 1 in the previous section, except the value of the intercept parameter a of the demand function, which is normalised to unit without any further loss of generality. The consequences of the choice of the regulatory regime for consumers’ surplus are illustrated in the following Proposition.
Proposition 4. Let VA(K) and VR(K) be the equilibrium aggregate con- sumers’ surplus under the Absolute and the Relative regime respectively.
Then, i) if f (K) < φA(K), then VR< VA; ii) if φA(K) < f (K) < φR(K), then VA< VR.
7All the numerical simulations are carried out using routines in the mathematical software Maple. These routines and the full results are available from the authors upon request.
This Proposition claims that, whenever entry occurs only under the Rel- ative regime, this is also the regime that brings about the highest level of aggregate consumers’ surplus. This happens because without entry the regulated firm sets the same price in both markets; the symmetry of the demand functions across the two markets, the quasi-convexity of consumer surplus and the slope equal to -1 of the average price cap constraint imply that consumers always prefer to a pair of identical prices any pair of prices along the same price cap constraint and, a fortiori, any price pair which lies below the average price cap line. Hence, the Relative regime turns out to be preferred since, by fostering entry that would not occur under the alternative regime, it pushes down the competitive price but also induces the firm to charge different prices in the two markets. On the other hand, when entry occurs under both regimes, consumers on the whole are better off under the Absolute regime rather than under the Relative one. When K is not ’too’ high, this is simply the effect of the Relative regime causing a higher competitive price, being the captive price identical across regimes;
however, consumers prefer the Absolute regime even for high values of K, since the reduction in the monopoly price brought about by this regime is not able to outplay the negative effects of the higher competitive price.
We now turn to assess the effects of the two regimes on aggregate profits.
In order to do so, we specify the nature of the entry cost borne by the po- tential entrant. For the sake of simplicity, we hypothesise a linear functional form, that is f (k) = tk. Then, denoting with tAand tRthe highest values of t for which entry occurs under the Absolute and Relative regime respectively, it is easy to show that tA = 1+c−K2 ; also, tR = β+11 when K00 < K < K000 while tR = (β+1)(1+c)−K
2(β2+1) when K000 < K < KmaxR . Consistently with the result illustrated in Proposition 3, it is easy to ascertain that tA< tR, which confirms that entry is more likely to occur under the Relative regime. We can now move on to compare the effects of the different regulatory regimes on aggregate profits. We find some results similar in nature to the pattern exhibited by aggregate consumers’ surplus, but with an opposite sign. These are illustrated in the following Proposition.
Proposition 5. Let ΠA(K) and ΠR(K) be the equilibrium aggregate indus- try profits under the Absolute and the Relative regime respectively. Then, i) if t ≤ tA, then ΠA< ΠR; ii) if tA≤ t ≤ tR, then ΠR< ΠA.
This Proposition basically says that when entry occurs only if the Rela-
K c β t VA− VR ΠA− ΠR WA− WR a) 0.505 0.125 1.143 0.350 -0.0615 0.1363 0.0748 b.1) 0.296 0.125 1.143 0.117 0.0204 -0.0105 0.0099 0.609 0.125 1.143 0.117 0.0453 -0.0506 -0.0053 b.2) 0.385 0.062 1.154 0.118 0.0114 -0.0183 -0.0069 0.385 0.187 1.154 0.118 0.0357 -0.0203 0.0154 b.3) 0.440 0.125 1.071 0.210 0.0362 -0.0272 0.0091 0.440 0.125 1.286 0.210 0.0160 –0.0204 0.0045
Table 1: Welfare differences across the regulatory regimes in Example 1
tive regime is in place, this regime delivers lower aggregate profits. This is because the aggregate profits obtained under this regime by the two firms operating as duopolists are clearly lower than the profits accruing to the regulated firm acting as the sole supplier under the Absolute regime. On the other hand, when entry occurs irrespective of the regime, the Relative regime delivers higher aggregate profits. This is because the higher com- petitive price under this regime clearly makes the potential entrant better off, and has a positive ’direct effect’ on firm M’s profits. This outplays the opposite effect on firm M’s profits due to the lower captive price when K is sufficiently large.
Propositions 4 and 5 together imply that the welfare comparison between the two regimes crucially depends on the distributional concern of the reg- ulator, that is on the size of parameter α. The highly non-linear nature of the welfare function once evaluated at equilibrium prices prevents us from fully characterising the equilibrium choice of the regulator. Restricting our analysis to the case of α = 1, we compare the two regimes by means of a numerical simulation which allows to evaluate the equilibrium welfare val- ues under the two regimes for a fine grid of admissible values of the model’s parameters.8
Some of the results of our analysis are reported in Table 1. When entry occurs only under the Relative regime, the numerical analysis shows that this is also the socially preferred regime. This means that the benefits accruing to consumers because of the more competitive environment under the Relative regime outweigh the loss in social welfare due to the duplication
8Clearly, Proposition 4 illustrates also the ranking of the two regimes in terms of social
of entry cost. Notice that this statement holds true even when firm E is less efficient than the rival, as it results from a value of t higher than c (see the section a) in Table 1).
The other sections of Table 1 give examples of the welfare comparisons between the two regimes when entry is profitable under both regulatory rules. Example b.1) shows that the positive effect on profits stemming from the Relative regime outplays the negative effects on consumers’ surplus. This is due to the fact that, when K is sufficiently large, the entrant profits are much larger under the Relative regime (because of the less aggressive price response by the incumbent) and able to counterbalance the lower incum- bent’s profits and the lower consumers surplus associated with this regime.
On the other hand, section b.2) shows that the lower is the unit cost for firm M, the more likely is that the Relative regime is welfare superior. This derives from the feature of the Relative regime to transfer to the monopo- listic market the price reductions obtained in the competitive market; the lower is c the greater is the positive effect on consumers’ surplus because of a reduction both in the competitive and in the captive price. Section b.3) illustrates that the lower is β the less likely is that the Relative regime is welfare superior. This is mostly due to the effect of this parameter on con- sumers’ surplus. Indeed, when β is low, the Relative regime forces firm M to choose prices very close the 45◦ degree line which, because of the symmetry of the convex isowelfare curves, are more likely to lie on a isowelfare which is lower than the one associated with prices set under the Absolute regime.
A final remark is on the role of the parameter t which only plays a role in the choice of entry but that, when entry occurs under both regimes, does not play any role in the welfare comparison between the two regimes.
Example 2: Linear demand and supply functions
The set-up is identical to the one of the previous example, except for firm E’s profits function for unit of capital. This is assumed to take the form e(p2) = 12(p2)2, which in turn implies that its supply function per unit of capital is linear and given by s(p2) = p2. This assumption reduces the impact of having posited that the capacity of the entrant is exogenously given and allows for a meaningful separation between the entry game and the pricing game. Due to this hypothesis on the supply function, entry will be more likely to enhance welfare with respect to the set-up with totally
K c β t VA− VR ΠA− ΠR WA− WR a) 1,480 0,160 1,050 0,040 0,1435 -0,1419 0,0016
2,030 0,160 1,050 0,040 0,1907 -0,1920 -0,0013 b.1) 0,390 0,110 1,100 0,020 -0,0029 0,0131 0,0102
1,390 0,110 1,100 0,020 0,0112 -0,0230 -0,0117 b.2) 0,570 0,010 1,000 0,020 0,0057 -0,0133 -0,0076 0,570 0,210 1,000 0,020 -0,0094 0,0353 0,0259 b.3) 0,810 0,110 1,050 0,020 0,0026 0,0015 0,0041 0,810 0,110 1,200 0,020 -0,0033 -0,0014 -0,0046
Table 2: Welfare differences across the regulatory regimes in Example 2
inelastic supply. Indeed, the larger elasticity of supply and of the residual demand curve facing the incumbent implies that the (positive) effects of entry on equilibrium prices and welfare are more substantial.
Equilibrium prices under the two regulatory rules closely mimic those ob- tained in the case of unit-elastic supply and are not reported here.9 Clearly, also the result regarding the higher likelihood of entry under the Relative regime is confirmed. Because of the complexity of the equilibrium welfare measures, we once again resort to numerical methods. The analysis is carried out varying key parameters and variables (c, β and K) within their admis- sible ranges and comparing equilibrium welfare values under the different regimes.
As in Example 1, we find that, when entry occurs only under the Relative regime, consumers prefer this regime where aggregate profits are also lower.
However, differently from the results obtained in Example 1, the effect on aggregate welfare is ambiguous, as it is shown in section a) of Table 2. This is due to the fact that, because of the elastic supply function and the more elastic residual demand function, a large K generates now a negative effect on aggregate profits which is larger than the one obtained in the case of inelastic supply and which outplays the positive effect of consumers’ surplus.
When entry occurs irrespective to the regime in force, the effects of the different parameters on consumers surplus and aggregate profits are similar to those observed in the case of inelastic supply and the discussion provided there applies. However, the magnitude of these effects is now larger than in
9The main difference with respect to the inelastic supply case is that prices are not linear in K. All the details are available from the authors upon request.
the previous case so that we find that the overall effect of a change in one of the parameters may generate a change in the differences given in the last three columns of the table which is not only of size but also of sign.
Consider first the effects of the two regimes on consumers’ surplus. We find that the Absolute regime brings about a higher aggregate consumers’
surplus provided K is sufficiently small. Instead, when K is large enough, the ranking between the two regimes is reversed, contradicting the findings for the case of inelastic supply (see section b.1) of Table 2). Intuitively, the more competitive environment faced by firm M in the present set-up places a stronger downward pressure on the competitive price. When entry occurs at a sufficiently large scale - so that this effect is quite substantial -, this effect and the induced reduction of the captive price occurring under the Relative regime makes it preferred by consumers. Similarly, provided that K is sufficiently large, aggregate profits under the Absolute regime are higher than those under the Relative regime (see again section b.1) of Table 2), differently from the case of inelastic supply. When K is sufficiently small, the ranking between the two regimes is reversed. Indeed, together with the forces already described in the case of inelastic supply, there is now one more factor affecting aggregate profits. Since the entrant supply function is now price elastic, the Relative regime, by causing a higher competitive price, has a stronger negative effect on firm M’s profits because of a ’business stealing’ effect. This effect becomes predominant when the scale of entry of the entrant is sufficiently large. Sections b.2) and b.3) in Table 2 focus on the consequences on varying the parameters c and β. The qualitative effects on social welfare are as discussed in Example 1; however, in the example at hand, lowering c or increasing β may reverse the welfare ranking between the two regulatory rules.
Considering now aggregate social welfare, we find that, when entry oc- curs only under the Relative regime, this regime leads to a higher aggregate social welfare only when K is sufficiently large. This is different from what we find in the case of inelastic supply when the Relative regime was always socially more desirable. The reason of the difference is again the more com- petitive environment under analysis here - due to the more elastic residual demand faced by the incumbent -, which makes consumers benefit more from entry when this occurs at sufficiently large scale. Lastly, in the case of entry under both regimes, the results of our analysis are qualitatively identical to
the findings for the case of inelastic supply.
5 Conclusions
This paper has analysed the effects of different ways of limiting the pricing discretion of a price capped firm which faces different degrees of competition in the markets it serves. This issue is of a particular importance when the regulated firm is price capped, since it has been shown in the literature that the firm can exploit its freedom in setting prices to hinder the development of competition. This issue is also relevant in more advanced stages of a liberalisation process, when price constraints in the relatively competitive markets are completely abolished; in this case, the key issue is whether the level of the remaining price constraints should be linked in some way to the prices prevailing in the more competitive markets.
The main policy implication that can be derived from this paper is that the use of cross-market restrictions may be part of a regulatory policy pro- moting competition. This type of constraints are more likely to enhance consumers’ welfare (and reduce aggregate firms’ profits) the higher is the degree of competition of the markets where entry has occurred - as approx- imated by the scale of entry of the potential entrant and by its elasticity of supply.
The strength of this result could clearly be enhanced if one took into account the dynamic advantages of a more competitive market. Moreover, even in the present static set-up, we argue that the advantages deriving from the cross-market price restrictions highlighted in our model are probably undervalued because of our hypothesis of price taking behaviour of the en- trant firm. Indeed, a model which allows for a strategic interaction between firms would provide the regulated firm with higher competitive pressure, thereby enhancing the likelihood of cross-market price restrictions to make consumers better off.
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A Appendix
Proof of Proposition 1. Let phi = phi(K) and pbhi = phi( bK) for i = 1, 2, h = A, R and K, bK > 0. By definition, π1(ph1) + π2(ph2) − Ks(ph2)(ph2 − c) ≥ π1(ph1) + π2(ph2) − bKs(ph2)(ph2 − c), and π1(pbh1) + π2(pbh2) − bKs(pbh2)(pbh2− c) ≥ π1(bph1) + π2(pbh2) − Ks(pbh2)(pbh2 − c). Adding up the two inequalities yields ( bK − K)s(ph2)(ph2− c) ≥ ( bK − K)s(pbh2)(pbh2− c). Since p2≥ c and s(p)(p − c) is increasing in p, ( bK − K) has always the opposite sign to (pbh2 − ph2).
Proof of Proposition 2. Let p02be the price which solves ϕ(p2) = b(p2). From the facts that i) b(.) is strictly increasing; ii) ϕ(.) is strictly decreasing, and iii) p1 is constant and equal to p01, i.e. the price which solves w. r. to p1 the equation p = β(ϕ(p)), it follows that b(p2) < (>) ϕ(p2) and p1 < (>
) ϕ(p2) if and only if p2 < (>) p02. From the definition of the regulatory regimes and the hypothesis that the price cap never allows to charge the unconstrained profit-maximising price, it follows that pA1 = min{ϕ(pA2), p1} and pR1 = min{ϕ(pR2), b(pR2)}. According to which expression is lower in each of the two prices, 4 different cases may occur:
1. ϕ(pA2) ≤ p1and ϕ(pR2) ≤ b(pR2) imply pA1 = ϕ(pA2) and pR1 = ϕ(pR2) and pA2, pR2 ≥ p02. When the first two weak inequalities hold as equalities, then it must be the case that pA2 = pR2 = p02. Hence, also pA1 = pR1 = p1. When the two inequalities hold as strict inequalities, then it must also be the case that pA2, pR2 > p02. However, since only the price cap constraint is binding under both regimes and this constraint is identical across regimes, the solutions to the maximisation problems faced by the firms under the different regimes must be identical. Hence, pA1 = pR1 and pA2 = pR2. When ϕ(pA2) < p1and ϕ(pR2) = b(pR2), it follows that pA2 > p02 and pR2 = p02 which, by a revealed preference argument, it can be easily shown to be inconsistent with profit maximising behaviour.
The same reasoning applies to the case of ϕ(pA2) = p1 and ϕ(pR2) <
b(pR2).
2. ϕ(pA2) ≤ p1 and ϕ(pR2) ≥ b(pR2) imply pA1 = ϕ(pA2) and pR1 = b(pR2) and pR2 ≤ p02 ≤ pA2. When the two first weak inequalities hold as equalities, then we are in a situation which is identical to case 1. When at least one of the two weak inequalities holds as a strict inequality,
by a revealed preference argument, prices can be easily shown to be inconsistent with profit maximising behaviour.
3. ϕ(pA2) ≥ p1 and ϕ(pR2) ≤ b(pR2) imply pA1 = p1 and pR1 = ϕ(pR2), but also pA1 ≥ pR1 and pA2 ≤ p02 ≤ pR2. Also in this case, when the first two weak inequalities hold as equalities, we are in a situation which is identical to case 1. When the two inequalities hold as strict inequalities, pA1 > pR1 and pA2 < p02 < pR2. On the other hand, when ϕ(pA2) > p1 and ϕ(pR2) = b(pR2), then pA1 = pR1 = p1 and pA2 < p02= pR2. Finally, the case of ϕ(pA2) = p1and ϕ(pR2) < b(pR2) is again inconsistent with profit maximising behaviour.
4. ϕ(pA2) ≥ p1 and ϕ(pR2) ≥ b(pR2) imply that pA1 = p1 and pR1 = b(pR2), which immediately establishes that pA1 ≥ pR1. Also in this case, when the two weak inequalities hold as equalities, we are in a situation which is identical to case 1. When the two inequalities hold as strict inequal- ities, it must be the case that pA2 < pR2. To show this, notice that under the Absolute regime only the p1 constraint is binding: hence, pA2 = arg max(π(p1, p2)). Since π(p1, p2) is additively separable in prices, pA2 does not depend on p1. Hence, since under the Relative regime only the b(.) constraint is binding, conditions which identify the optimal prices under the two regimes can be written as follows:
∂π
∂p1
∂π
∂p2
pi=pAi
= dpdp1
2 = 0 and
∂π
∂p1
∂π
∂p2
pi=pRi
= dpdb
2 > 0, for i = 1, 2. Since the slope of the profit contour can be positive only when p2 is above its unconstrained level (provided that p1 is below its unconstrained level), this establishes the result. Finally, the cases in which only one of the inequalities holds as a strict inequality while the other holds as an equality have already been dealt with in cases 2 and 3.
Proof of Corollary 1. From the symmetry of the two markets and of ϕ(.), when k = 0, pA1 = pR1 = pA2 = pR2 but also ph1 = ϕ(ph2) for h = A, R. From Propositions 1 and 2, pA2|k=K ≤ pR2|k=K < pA2|k=0 = pR2|k=0. Hence, there must exist a threshold value of eK ≥ 0, such that for any K ≤ eK, pA2 = pR2. From the proof of Proposition 2, this also implies that pA1 = pR1.
Proof of Proposition 3. Trivial, by simply combining the fact that pA2 ≤ pR2
