Pro-Consumer Price Ceilings under Uncertainty

Transcription

1 Pro-Consumer Price Ceilings under Uncertainty John Bennett y Ioana Chioveanu z March 11, 2015 Abstract We examine ro-consumer rice ceilings under regulatory uncertainty about demand and suly. In a erfectly cometitive benchmark with demand uncertainty only, the otimal ceiling always binds if uncertainty is small. If uncertainty is large, it may bind, deending on the realized demand. These ndings are robust in imerfectly cometitive settings where symmetric rms comete in suly functions. Moreover, the otimal ceiling is weakly increasing in the degree of cometition. We also examine rationing ine ciency and in down a cut-o level of rationing e ciency above which the ndings in our benchmark model are robust. Similar qualitative results obtain with uncertainty regarding both demand and suly and we illustrate this solution numerically. Keywords: rice regulation, consumer surlus, uncertainty JEL classi cation: L50, D82, D41, L13, D45 We thank U¼gur Akgün, Helmuts Azacis, Jan Fidrmuc, Pablo Guillén, Elisabetta Iossa, Manfredi La Manna, James Maw, and Matthew Rablen for useful comments. The usual disclaimer alies. y Deartment of Economics and Finance, Brunel University, Uxbridge UB8 3PH, UK. john.bennettbrunel.ac.uk. z Deartment of Economics and Finance, Brunel University, Uxbridge UB8 3PH, UK. ioana.chioveanubrunel.ac.uk. 1

2 1 Introduction Although controversial, rice ceilings have been used in a wide range of markets, including those for rental accommodation, gas, electricity, telecommunications, and ay-day lending. One rationale for introducing a rice ceiling is to correct market ine ciency stemming from insu cient cometition. Another is the rotection of consumers, which may exlain the use of rent controls in markets that are nearly cometitive. Insofar as rice regulation is designed to correct ine ciencies from market failures, it would aim to maximize total welfare. Nonetheless, emirical evidence on ast intervention suggests that the realities of regulation are more comlex, with lobby grous in uencing regulatory intervention in their own interests. 1 Recent years have witnessed an increase in the focus of olicy makers and regulators on consumer welfare. 2 debate. 3 This has coincided with an increased role of consumer grous in olicy In the UK, following the 2002 Enterrise Act, aroved bodies were designated suer-comlainants at the O ce of Fair Trading (now art of the Cometition and Markets Authority). This was exlicitly done to strengthen the voice of consumers and rotect their interests. 4 analysis. 5 These changes motivate the study of a consumer-surlus standard in economic A major roblem faced by a regulator when setting a rice ceiling is uncertainty over demand and suly conditions. 6 Yet, although a small literature analyzes the imact of rice ceilings on consumers (e.g., Davis and Kilian, 2011, Bulow and Klemerer, 2012), there is no study that exlores otimal ro-consumer ceilings under regulatory uncertainty. We develo a model of rice regulation where the regulator is imerfectly informed about demand and suly and is aware of this informational disadvantage. However, rivate agents have all relevant information, so that uncertainty is entirely on the art of the regulator. In this 1 For instance, in this context, Viscusi et al. (2005, Ch.10) rovide examles of intervention that raised industry ro ts. 2 Possible motives for these changes include the deregulation of utility markets, increased market comlexity (e.g., banking and nancial services), and growing evidence on consumer bounded rationality. 3 Traditionally, industry lobbying (small sulier grous) has been more e cacious. Consumers (a large but fragmented grou) have been less e ective due to the large ga between aggregate and individual gains. 4 See O ce of Fair Trading 511 (2003). The consumer organization Which? was one of the rst bodies to be given this designation and it recently in uenced tari regulation in the energy sector. 5 This standard has been used in merger evaluations in Euroe and the US. See Besanko and Sulber (1993) and Neven and Röller (2005) for related analyses, and Motta (2004) for a discussion including caveats of this standard. 6 For examle, a regulator may not have detailed knowledge of what innovations are in the ieline or of the demand for new roducts. 2

3 sense, demand and suly can be regarded as being subjectively random. 7 We assume that the regulator s objective is to maximize exected consumer surlus, and derive conditions under which intervention may be desirable. Our framework allows for varying degrees of cometition, so it ts a gamut of market structures. We rst resent a reliminary analysis of a erfectly-cometitive market with e cient rationing and show that with no uncertainty the rice ceiling that maximizes consumer surlus always binds. The reduction in rice due to regulation increases the surlus of consumers who still urchase, whilst the resulting dro in quantity sulied decreases it; the rice e ect revails and the net imact on consumer surlus is ositive. We further show that this result is robust to the introduction of relatively low levels of demand (or suly) uncertainty. We then introduce arbitrary levels of regulatory uncertainty regarding demand in this erfectly cometitive benchmark. We characterize the otimal rice ceiling and show that, with su ciently small uncertainty, it binds for any resolution of the uncertainty. In contrast, if uncertainty is large enough, the otimal rice ceiling is set at a higher level and may bind or not, deending on the demand realization. In this case, we derive conditions under which the otimal ceiling lies above or below the exected market-clearing rice. 8 To understand how the interlay between demand uncertainty and the degree of cometition a ects the otimal ceiling, we emloy an imerfectly cometitive framework where identical rms comete in linear suly functions. We draw on the fact that in this setting aggregate suly in the imerfectly cometitive equilibrium is a fraction of the erfectly cometitive one. 9 We show that our qualitative results in the benchmark model are robust to this generalization. More seci cally, with little uncertainty, the regulator sets a low otimal ceiling that surely binds and is indeendent of the degree of cometition. For large regulatory uncertainty, the otimal ceiling may bind or not deending on the realized demand. Furthermore, we nd that the otimal ceiling is weakly increasing in the degree of cometition. Intuitively, when demand and suly are deterministic (or nearly so), the otimal ro- 7 This information structure also covers the ossibility that the rice ceiling is a long-term regulatory decision, whereas short-run market conditions may change. It has been widely emloyed in the theory of regulation (see Armstrong and Saington, 2007) and was rst formalized by Weitzman (1974) and subsequently used in analyses of regulation starting with Baron and Myerson (1982) and Lewis and Saington (1988). 8 This benchmark model focuses on regulatory uncertainty regarding the demand. In an online aendix we show that qualitatively similar results obtain in a model with suly uncertainty only. In section 6 we analyze two-sided uncertainty. 9 The reduced form model with an aggregate outut restriction relative to the erfectly cometitive benchmark can also reresent a monooly and so our results aly to this limiting case. 3

4 consumer ceiling is strictly below the market clearing rice and, imlicitly, the imerfectly cometitive rice. cometitive ressure. Furthermore, this otimal ro-consumer ceiling is indeendent of the However, when there is relatively large uncertainty the imact of the ceiling will deend on the degree of cometition, as that determines the rice in the unregulated market. ln addition, the exected consumer surlus loss in the unregulated market is larger in a more concentrated market which creates a stronger incentive to lower the rice ceiling. This triggers the ositive relationshi between the otimal ceiling and the degree of cometition. Using variants of the erfectly-cometitive benchmark model, we further examine the robustness of our ndings. First, we analyze the interaction of rationing ine ciency with the level of demand uncertainty in the determination of the otimal ro-consumer rice ceiling. We in down a cut-o level of rationing e ciency above which our ndings in the benchmark model with erfectly e cient rationing are qualitatively robust, and the otimal rice ceiling deends on the degree of uncertainty. In contrast, if rationing e ciency is below this cut-o level, regardless of the degree of uncertainty, it is otimal for the regulator to set a ceiling so high that it never binds. 10 At a binding rice ceiling all consumers who buy the roduct enjoy a higher surlus than they would in the absence of regulation. But with ine cient rationing some of these consumers dislace others who value the roduct more than they do. If rationing is su ciently ine cient, then the loss in consumer surlus due to such ine cient reallocations and lower suly can fully o set the aggregate bene ts of the consumers (who buy) at the lower rice. For examle, if the regulator knows that incumbents have a strong advantage in obtaining rental accommodation, although the needs of temorary residents are greater, then it should set a relatively high rent control. 11 Finally, we examine regulatory uncertainty regarding both demand and suly, while also allowing for rationing ine ciency. In this analysis, which draws on our results in the two olar cases with demand or suly uncertainty only, the regulator s objective function becomes very involved and the otimal rice regulation olicy cannot be fully characterized. However, we are able to show that our qualitative ndings in the cases where the uncertainty 10 Our model with suly uncertainty only, which we analyse in the online aendix, also allows for rationing ine ciency and exlores the interlay between this and the degree of uncertainty. The results we obtain are consistent with those with only demand uncertainty. 11 Thus, in ractice rationing may even be worse than random (see Glaeser and Luttmer (1997)). Allowing for this in our model, we nd that if the ine ciency of rationing is extreme, rice should be left uncontrolled, irresective of demand and suly sloes. 4

5 is for one side only are robust to this generalization and we use numerical simulations to investigate further the joint imact of demand and suly uncertainty. Price ceilings in the resence of uncertainty in a Cournot oligooly with homogeneous roducts were analyzed by Earle, Schmedders, and Tatur (2007). They show that some commonly-made arguments in favour of such intervention break down when there is demand uncertainty. In articular, they nd conditions under which, in contrast to the no-uncertainty case, a lower rice ca (above marginal cost) leads to lower outut. But, unlike ours, their setting, which is develoed further by Grimm and Zöttl (2010) amongst others, focuses on rms uncertainty regarding the imact of the rice ceiling. 12 We exlore the role of regulatory uncertainty and concentrate on intervention that aims to maximize exected consumer surlus. Bulow and Klemerer (2012) analyze theoretically the e ect of rice regulation on consumer surlus under certainty allowing for general demand and suly functions. They show that, with random rationing, if suly is locally more elastic than demand, a rice ceiling always hurts consumers when demand is convex. In a no-uncertainty variant of our model, with random rationing, even if demand and suly curves are known to the regulator, exected consumer-surlus maximization requires the regulator to leave the rice uncontrolled if the demand curve is steeer than the suly curve. This result is a seci c case of Bulow and Klemerer s ndings; they show that it is robust when rent-seeking dissiates surlus further. Other models of rice regulation and e ciency in cometitive markets by Glaeser and Luttmer (1997, 2003) and Davis and Kilian (2011) also stress the welfare loss associated with ine cient rationing. Glaeser and Luttmer develo a model to examine emirically the misallocation due to rent control and show that under conservative assumtions, 20% of rented aartments in New York City were in the wrong hands. Davis and Kilian (2011) analyze emirically the US residential market for natural gas and nd substantial allocative costs over the eriod. Our analysis contributes to this emerging literature by exloring the role of uncertainty and its interlay with rationing ine ciency. In addition, by arameterizing the e ciency of rationing, we roose a exible framework that allows for a range of outcomes and extends revious work. Section 2 formulates the model and section 3 resents a reliminary analysis of a erfectlycometitive benchmark with demand uncertainty to illustrate some basic ndings. In section 12 A similar information structure is used by Dobbs (2004), who analyses the imact of a rice ceiling on a monoolist s investment decisions. 5

6 4 we comlete this analysis by allowing for arbitrary levels of regulatory uncertainty. Section 5 analyzes a arallel model with imerfect cometition. Section 6 exlores the imlications of rationing ine ciency and of two-sided uncertainty (with both demand and suly stochastic) using our benchmark model. Section 7 resents the conclusions. All roofs missing from the text are relegated to aendices. 2 The Model Consider a market in which the regulator may be uncertain about the demand and suly for a homogeneous roduct. There is no uncertainty on the art of rivate agents. On this basis, the regulator chooses a rice ceiling that maximizes exected consumer surlus. The regulatory intervention is announced to all rivate agents (roducers and consumers). First we introduce a erfectly-cometitive benchmark model. Let consumers gross bene t (or utility) from the consumtion of q units of the roduct be given by B(q; ) = (B + )q bq 2 =2 where b > 0; B + > 0. From the regulator s oint of view is a random variable with zero mean (E() = 0) distributed according to a twice continuous and di erentiable c.d.f. F() de ned on a closed interval [n min ; n max ]. We assume that the hazard rate F 0 ()=(1 F()) is strictly increasing. 13 The inverse demand is P d (q; ) = B(q; )=q = B + bq : The suliers cost of roducing q units is given by C(q; ) = (C + )q + cq 2 =2 where c > 0; C + > 0. From the regulator s oint of view is a random variable with zero mean (E() = 0) distributed according to a twice continuous and di erentiable c.d.f. G() de ned on a closed interval [t min ; t max ] : We assume that the hazard rate G 0 ()=(1 G()) is strictly increasing. The two shocks and are assumed to be indeendent. The inverse suly is P s (q; ) = C(q; )=q = C + + cq : Writing for unit rice, it follows that direct demand and suly are given, resectively, by q d () = B + b and q s () = C c : (1) As the uncertainty in the model is entirely on the art of the regulator, the shocks and are subjectively random. The rivate agents observe their realized values before making 13 The hazard rate is also known as the death rate, for instance in actuarial science. With continuous time, the death rate gives the instantaneous robability of dying at time t conditional on having survived to t. It is therefore intuitive to assume that the hazard rate is increasing. This assumtion is satis ed by a wide range of c.d.f:s used in economics. 6

7 decisions. We assume that roducers can observe (as well as ), while the regulator cannot. This catures the informational advantage that roducers may have over the regulator regarding the demand. 14 Denote by the ex-ost market-clearing rice (where q d ( ) = q s ( )) and by q the corresonding outut level: Then, = c(b + ) + b(c + ) b + c and q = B + C b + c : (2) We assume that B + > C + for any and. This guarantees a well-de ned equilibrium outut ex ost. Note that ex ante (before the demand and suly shocks and are realized), the regulator views (; ) as a random variable with exected value e = cb + bc b + c : (3) We exlore regulatory intervention that takes the form of a rice ceiling, assuming that resale of rationed goods is not ossible. Throughout, for simlicity, we restrict attention to < B + n min, thereby ruling out the ossibility of zero demand. A rice ceiling stiulates a maximal trade rice and only binds if the unregulated market rice lies above the regulated level. If it lies at or below the ceiling, the outcome coincides with the unregulated market equilibrium; that is, for a given rice ceiling ; if, then q() = q (as given by (2)) and if >, then q() = q d () (as given by (1)). The c.d.f. of (; ) is determined by the c.d.f.s of and ; that is, F() and G(), resectively. Since both and are de ned on closed intervals, so is the c.d.f. of (; ). We then examine rice ceilings in a model of imerfect cometition, maintaining our assumtions regarding the demand side of the market and the quadratic cost resented above. We assume that there are n identical suliers and each rm s cost of roducing q i units is given by C(q i ; ) = (C + )q i + ncq 2 i =2. 15 Moreover, the rms comete by choosing linear suly functions. Building on Klemerer and Meyer (1989), each rm chooses a suly function S i ( C ) = d i ( C ), where the sloe d i > In the linear suly function 14 It may be that is unknown when the regulator sets a rice ceiling, but is revealed by the time the roducers make suly decisions; or it may be that roducers can adjust their behavior as information about is revealed by the market. 15 This individual cost function guarantees that when the total cost of roducing q using n-lants in the industry are minimised, total cost equal is C(q) = (C + )q + cq 2 =2, the same as in the erfectly cometitive benchmark: 16 Klemerer and Meyer (1989) rovide a thorough analysis of suly function cometition. In articular, they show that with unbounded demand uncertainty and a symmetric industry there is a unique equilibrium where rms choose linear suly functions of the form we consider. This result holds regardless of the distribution of uncertainty (even if it degenerates into a mass oint), so long as the suort is unbounded. See Weretka (2011) for further discussion. 7

8 equilibrium, a rm s suly is only a fraction of its suly in the cometitive benchmark (see Akgün (2004) for related analysis). Thus, with imerfect cometition, the rms restrict their outut comared to that in the erfectly cometitive market. It then follows that the aggregate quantity sulied in the symmetric suly function equilibrium at rice can be written as q s (; ) = C c where < 1 catures the restriction in outut below the cometitive level. It is straightforward that increases in the degree of cometition, catured by the number of rms, n. As the market becomes nearly cometitive (n! 1), converges to 1. We formally show these results for our framework in the aendix. Using q s (; ), we can derive the equilibrium outcome in the imerfectly cometitive market. Denote by the ex-ost unregulated market rice (where q d ( ) = q s ( ; )) and by q the corresonding outut level: Then,, = c(b + ) + b(c + ) b + c and q = (B + C ) b + c : As in the cometitive framework, the regulator views the unregulated market rice (; ) with imerfect cometition as a random variable. In general, regulation that takes the form of a rice ceiling may result in excess demand, in which case the scarce outut will be rationed. Although rationing may in rincile be e cient, with the suly of outut allocated to the consumers whose valuations of the good are highest, in ractice there may well be some degree of ine ciency. A recent literature (see, for instance, Davis and Kilian, 2011) has focused on the alternative of random rationing, whereby the available suly is allocated randomly amongst those consumers whose valuations of the roduct are at least as great as the sale rice. Intuitively, the scoe for ro-consumer rice regulation is limited by rationing ine ciency. To examine the imact of such ine ciency, we adat our cometitive benchmark by introducing a arameter 2 [0; 1] that catures the e ciency of rationing. We write the regulator s objective function as a linear combination of the exected consumer surlus for e cient rationing and for extremely ine cient rationing where the available suly is allocated to the consumers whose valuations are the lowest among those who are willing to buy. 17 In artic- 17 Alternatively, may be interreted as the robability, for the regulator, that rationing will be e cient. If this robability is indeendent of the other stochastic variables in the model, we obtain the solution for uncertainty of the rationing scheme. Another interretation is that the available suly is slit, with di erent ortions being rationed with di erent degrees of e ciency. 8

9 ular, rationing is e cient when = 1; extremely ine cient when = 0, and random when = 1= A Preliminary Analysis This section introduces some of our results in a textbook framework of erfect cometition and e cient rationing (i.e., we assume that = 1 and = 1). We start with a situation where demand and suly are deterministic, and show that consumer surlus can be increased from the free-market benchmark by setting an aroriate binding rice ceiling. 19 We then discuss why this result still holds when there is a limited amount of demand uncertainty. The next section analyzes formally the demand-uncertainty case, allowing for varying degrees of uncertainty. Using the model introduced in section 2, let us initially assume that = = 0: The demand and suly schedules become q d () = (B )=b and q s () = ( C)=c, and the equilibrium rice in the unregulated market is = (cb + bc)=(b + c). For any rice ceiling, the market rice is and consumer surlus is given by CS(q d ( )) = Bq d ( ) 1 2 b qd ( ) 2 q d ( ) = b(b C)2 2(b + c) 2. However, any rice ceiling < is binding so that outut is min[q d (); q s ()] = q s () = ( C)=c. With e cient rationing, consumer surlus is CS(q s ()) = Bq s () 1 2 b (qs ()) 2 q s () = ( C)[2c(B ) b( C)] 2c 2 CS L d. (4) Using CS(q d ( )) and CS L d we obtain the following result. Lemma 1 With no uncertainty, the rice ceiling ^ that maximizes consumer surlus under erfect cometition with e cient rationing is given by CS L d ^ = cb + (b + c)c b + 2c <. (5) Proof. (i) At a rice ceiling, dcs(q d ( ))=d = 0. (ii) When <, di erentiating w.r.t., the f.o.c. gives the value in the roosition. The s.o.c. is always satis ed. It is easy to check that the otimal ceiling is well de ned, i.e., lies in the interval (C; ). 18 Emirical evidence suggests that in ractice 1=2: while rationing may not be extremely e cient, it is not worse than random. See, for instance, Glaeser and Luttmer (1997). However, we also consider rationing that is worse than random ( < 1=2) and nd conditions under which regulation may be bene cial even in this case: 19 In this setting, the free-market outcome is also the welfare-maximizing one. However, our analysis exlores ro-consumer rice regulation, that is, intervention that aims to maximize consumer surlus rather than total welfare. 9

10 Setting a rice ceiling slightly lower than has a negative e ect on consumer surlus due to a reduction in suly and the resulting exclusion of some low-valuation consumers from the market. However, there is also a ositive e ect as the lower rice makes consumers who still urchase enjoy a higher surlus. Panel A in Figure 1 illustrates the trade-o between these e ects at the otimal rice ceiling ^. The consumer surlus lost due to the outut reduction is catured by the dotted triangle, while the gain in the surlus of consumers who still urchase is catured by the dotted rectangle. As illustrated in the gure, the latter gain more than o sets the loss due to undersuly. The otimal ro-consumer rice ceiling balances this trade-o and is strictly lower than the free-market equilibrium rice, and so it binds irresective of arameter values. ^ P s (q) P d (q) E ( ()) (n min ) E (^()) P s (q) P d (q; n max ) P d (q) P d (q; n min ) q A B Figure 1: Otimal Pro-Consumer Price Ceiling with E cient Rationing q Before introducing uncertainty, let us brie y discuss how the analysis above would be a ected by nonlinearity of demand and suly. For a given demand-suly intersection and given sloes of demand and suly at this intersection, the same qualitative conclusion holds if there is strict convexity or strict concavity of either curve. However, strict convexity results in a higher otimal ceiling, while strict concavity results in a lower one than in the linear case. Consider a strictly convex demand tangent to the straight line P d (q) in Figure 1A at the intersection with P s (q). The consumer surlus loss from the ceiling ^ is then larger than in the gure, but the gain is the same. Still, there are binding ceilings for which the net gain is ositive (e.g., a ceiling marginally below ). A similar argument holds if suly is strictly convex. Then the loss is the same, and the gain is smaller than in the linear case. These conclusions are reversed for strict concavity. 10

11 Let us now introduce a small amount of demand uncertainty into the model. We continue to assume that suly is certain. Panel B in Figure 1 illustrates the highest and lowest demand functions, P d (q; n max ) and P d (q; n min ), resectively. In this case, P d (q) is the exected demand and catures a situation where the realized value of is equal to the exectation of, E() = 0. For any realization of, consumer surlus is maximized at a rice ceiling ^() = c(b + ) + (b + c)c b + 2c < (). This follows immediately from relacing B with B + in Proosition 1. The exected otimal ceiling is the same as the one resented in the roosition as E (^()) = ^. This is because the objective function only deends on the exectation of, E() = 0. This simle reasoning is correct so long as ^ < (n min ) - which is the case if there is only relatively little uncertainty. Therefore, our result from the certainty benchmark carries over to a model with relatively little uncertainty; the rice ceiling that maximizes exected consumer surlus binds regardless of the realization of demand. In contrast, if ^ > (n min ), the analysis will be di erent: such a ceiling may bind for some realizations of demand but not for others. 4 The Benchmark Model We now fully investigate the imact of arbitrary levels of demand uncertainty assuming suly is deterministic ( = 0). The analysis focuses on rice regulation where the ceiling satis es > C, so that suly is ositive. For a given rice ceiling, we de ne n (), the seci c value of for which the market clears, i.e. = (). Using (2) with = 0, we obtain n () = (b + c)( e) c, (6) where e = (cb + bc)=(b + c) is the exectation of (). In the arameter region where < (n min ) (i.e., for so that n () < n min ) a rice ceiling binds and results in excess demand for all values of. In the region where > (n max ) (i.e., for so that n () > n max ), a rice ceiling does not bind and the free-market outcome revails for all values of. However, in the region where 2 [ (n min ); (n max )] (i.e., for so that n () 2 [n min ; n max ]), the e ect of a rice ceiling deends on the value of. In articular, for low demand (when 2 [n min ; n ())) the rice ceiling does not bind, whereas for high demand (when 2 [n (); n max ]) it results in excess demand. Thus, we identify three regions 11

12 of rice ceilings where the intervention has di ering imlications. We will examine each of these three regions as otential locations for the otimal value of. I In the low rice region, regardless of ; the intervention binds. I In the middle region, the ceiling may bind or not deending on the realization of. I In the high rice region, the intervention does not bind. The low rice region: In this case < (n min ) and, regardless of ; suly is a binding constraint, so outut is min[q d (); q s ()] = q s () = ( C)=c and there is rationing. Consumer surlus is CS(q s (); ) = ( C)[2c(B+ ) b( C)]=2c 2, and exected consumer surlus is E(q s (); ) = CS L d roblem in this region is ^ in (5): given in (4). The rice ceiling that solves the f.o.c. of the otimization Before roceeding, we introduce some notation. De nition 1 Let n 0 = c(b C) b + 2c. If n min n 0, CS L d is increasing for all < (n min ) and the critical value ^ is weakly larger than (n min ), which is inconsistent with the region (C; (n min )): However, if n min > n 0, as d 2 CS L d =d2 < 0, (5) is a well-de ned local maximum within the region (C; (n min )). This roves the following result. Lemma 2 With demand uncertainty only, if n min > n 0, then in the region where < (n min ) the rice ceiling that maximizes exected consumer surlus under erfect cometition is given by (5). If n min n 0, an otimal rice ceiling cannot be strictly lower than (n min ). This result shows that with large enough demand uncertainty, in the sense that n min n 0, the otimal rice ceiling must be at least as high as (n min ) = min (): Otherwise, Lemma 2 shows that there is a local maximum that is lower than (n min ): After analyzing the other ossible levels of the rice ceiling, we will exlore the global otimality of this ceiling. The high rice region. Consider now a rice ceiling in the region > (n max ): In this case, regardless of the realization of ; the intervention does not bind. For a given, consumer surlus is b(b + and is the same as in the free-market equilibrium. C) 2 =2(b + c) 2 : So exected consumer surlus becomes b[(b C) 2 + E( 2 )] 2(b + c) 2 CS H d, (7) The middle rice region. We exlore rice ceilings in the region [ (n min ); (n max )], so that the corresonding n () 2 [n min ; n max ]: We make use of the following observation. 12

13 Remark 1 From the de nitions of () and n (), it follows directly that rob( = rob( n ()) = F(n ()). ) For any rice ceiling () (or, equivalently, n ()); the intervention does not bind, the market-clearing rice () revails and the quantity traded is q d ( ; ) = q s ( ). Since q d ( ; ) = B + b consumer surlus in this case is given by = B + C b + c q d ; CS(q d) = b(b + C)2 2(b + c) 2 : Using Remark 1, it follows that exected consumer surlus conditional on () is R E(CS(qd ) j n n ()) = () n min CS(qd )df() =F(n ()). Substituting CS(qd ), we obtain where L d (n ()) = R n () n min b (B C) 2 F(n ()) + 2(B C) L d (n ()) + L d (n ()) 2(b + c) 2 F(n ()) df() and L d (n ()) = R n () 2 df(). n min CS d ; For () (or, equivalently, n ()), however, the intervention leads to excess demand. The realized consumer surlus from the q s () = ( C)=c units roduced is the same as in the low rice region. Therefore, exected consumer surlus conditional on R () is E (CS(q s ()) j n nmax ()) = n () CS(qs (); )df() = [1 F(n ())], which becomes ( C) [2c(B ) b( C)] + ( C) H d (n ()) 2c 2 c(1 F(n ())) where H d (n ()) = R n max n () df(): CS S d, (8) Using CSd and CSS d, it follows that total exected consumer surlus for any given rice ceiling 2 [ (n min ); (n max )] is E(CS()) = F(n ())CS d + (1 F(n ())CS S d CS d : (9) We can now state the following result. Lemma 3 With demand uncertainty only, exected consumer surlus under erfect cometition with e cient rationing is continuous and di erentiable for all values of > C: Moreover, at any rice ceiling ((n max )) exected consumer surlus is indeendent of. 13

14 For 2 [ (n min ); (n max )]; the regulator sets the rice ceiling to maximize CS d : An interior otimal rice ceiling in this region solves dcs d =d = 0. Using (6), we di erentiate this exression with resect to and obtain dcs d d = 1 c L d (n ()) (b + 2c)( C) c(b C) c 2 (1 F(n ()) : (10) The roof of the next result is resented in the aendix. Proosition 1 With demand uncertainty only, if n min n 0, the unique rice ceiling that maximizes exected consumer surlus under erfect cometition lies in the interval [ (n min ); (n max )). In addition, this ceiling may be greater or smaller than the exected market-clearing rice. Seci cally, CS d is maximized at T e as (b + c) L d ( e) c(b C)(1 F(0)) T 0. Proosition 1 holds if the regulator faces enough uncertainty. This favours setting a rice ceiling that is relatively high and, therefore, less likely to bind. If the rice ceiling binds ex ost so that suly is a constraint, variation of the demand curve has no e ect on consumtion. However, if ex ost the rice ceiling does not bind, greater demand uncertainty generates a larger exected consumer surlus. Consumer surlus for a given equals (b=2) [(B + C)=(b + c)] 2 and so demand uncertainty ( 6= 0) raises the exected value of this exression. Since demand uncertainty has a ositive e ect on exected consumer surlus if the ceiling does not bind, and no e ect otherwise, it suorts a high rice ceiling that is less likely to bind. This works against the incentive to set a relatively low rice that we identi ed in our reliminary analysis and may even o set it. Examle 1 Suose is uniformly distributed on [ n; n] so that L d (0) = n=4 and F(0) = 1=2: The condition n min n 0 in Proosition 1 becomes n c(b C)=(b + 2c). It follows that CS d is maximized at T e as n T 2c(B C)=(b + c). Then, for n 2 [c(b C)=(b + 2c); 2c(B C)=(b+c)), the otimal rice ceiling is lower than e and for n > 2c(B C)=(b+c), the otimal ceiling is higher than e. The result that with relatively little uncertainty CS d is maximized at < e is related to the e ects discussed in section 3. There we saw that when demand and suly are certain, a rice reduction below the market equilibrium raises consumer surlus er unit bought. In contrast, when demand is stochastic, it is not known ex ante whether realized suly will be a binding constraint on consumtion and, as a result, we show that for large enough uncertainty CS d is maximized at > e. However, when there is small enough uncertainty, the result from the certainty case still holds so that CS d is maximized at < e. Moreover, with very little uncertainty, that is, if n min > n 0, the analysis receding Proosition 1 shows that an otimal rice ceiling cannot lie 14

15 in the interval [ (n min ); (n max )]: Together with the ndings in Lemma 3, this establishes the following result. Proosition 2 With demand uncertainty only, if n min > n 0, the unique rice ceiling that globally maximizes exected consumer surlus under erfect cometition is given by (5) and always binds. With relatively little uncertainty (n min > n 0 ), the otimal ro-consumer rice ceiling is strictly lower than (n min ) and, imlicitly, than the exected market-clearing rice. This result, along with Proosition 1, highlights the e ects of the degree of uncertainty on otimal regulation. Proosition 2 generalizes the initial analysis illustrated in Figure 1B. In contrast, if there is enough uncertainty (n min n 0 ), the otimal rice ceiling is higher than (n min ) as shown in Proosition 1. Although we focus on the role of demand uncertainty here, the analysis of the otimal rice ceiling in a setting with no uncertainty is a secial case. With no uncertainty, has a degenerate distribution so that n min = n max = 0. Since n min = 0 > n 0 the results in Lemma 2 aly, and the otimal rice ceiling is ^, as given by (5). Thus, the otimal rice ceiling is ^ with no uncertainty and remains at this level when a limited amount of uncertainty (in the sense that n min > n 0 ) is introduced. But we have seen that with greater uncertainty (for n min n 0 ) a rice ceiling in the middle region, 2 [ (n min ); (n max )], is otimal. In this case, the rice ^ belongs to the middle region. Evaluating (10) at ^, we obtain dcs d (^) d = 1 c L d (n (^)) 0 ; where the weak inequality follows from L d (n (^) 0, and holds with equality only for n min = n 0 (or, (n min ) = ^). Therefore, if uncertainty is great enough (n min n 0 ), the otimal rice ceiling which lies in the middle region is strictly higher than ^, the otimal ceiling with smaller or no uncertainty. 5 Imerfect Cometition and Demand Uncertainty We now analyze the rice ceiling that maximizes exected consumer surlus in the imerfectly cometitive framework resented in section 2. We assume that the regulator only faces uncertainty regarding demand ( is stochastic and = 0). We exlore the robustness of our ndings in the erfectly cometitive benchmark to a more realistic setting where suliers have (some) market ower, and analyze how changes in the degree of cometition a ect the otimal ro-consumer ceiling. In this model, < 1 measures cometitive ressure, with a 15

16 larger value corresonding to more intense cometition. This analysis builds on the suly function cometition microfoundation introduced in section 2 and it relates to environments that are imerfectly cometitive. However, at the end of this section, we show that our analysis is informative also for monooly markets. All derivations and roofs missing from the text are relegated to the online aendix. First note that in a model with no uncertainty, the otimal ro-consumer rice ceiling from our reliminary analysis still obtains with imerfect cometition because consumer surlus is maximized at the same rice level. However, as we show below, this is no longer true with large uncertainty. For a given realization of demand, the equilibrium rice in the unregulated imerfectly cometitive market is given by (; ) = c(b + ) + bc b + c and (; ) > () (where () is the market-clearing rice). We rst analyze a situation where (n min ; ) < (n max ), which is arguably more general. 20 For a given, (n min ; ) is the ex-ost equilibrium rice for the lowest demand realization, while (n max ) is the market clearing rice for the largest demand realization. As before, we focus on ro-consumer regulation that takes the form of a rice ceiling < B + n min. Whether binds or not deends on its osition relative to the imerfectly cometitive market rice (; ). When it binds, whether demand or suly will be the constraint deends on the osition of relative to the market clearing rice (): More recisely, if > (; ) then the intervention does not bind and the imerfectly cometitive market outcome revails. In contrast, if < (; ), the ceiling binds and the quantity traded in the market at the regulated rice is minfq d (); q s (; )g. Seci cally, if () < < () the quantity traded will be q d () (there is excess suly), while if < (), it will be q s (; ) (there is excess demand). (i) Consider rst a rice ceiling > (n max ). For values of at which the ceiling binds, demand is a constraint, that is minfq d (); q s (; )g = q d (). As a result, a small decrease in will lead to an increase in consumer surlus. At values of where the ceiling does not bind, a decrease in the ceiling either has no e ect on consumer surlus or, if it has, it leads to greater consumer surlus as the unregulated market rice is above the cometitive level for < 1. This shows that an otimal rice ceiling cannot be strictly larger than (n max ) 20 Even if for some values of, (n min ; ) (n max ), there exists a cut-o value 0, so that for 2 [ 0 ; 1), (n min ; ) < (n max ). This is because lim!1 (n min ; ) = (n min ) < (n max ). When! 1, the market becomes almost erfectly cometitive. The cut-o value 0 is imlicitly de ned by (n max ) = (n min ; 0 ). Towards the end of this section, we will return to the case where (n min ; ) (n max ). 16

17 as a slightly lower one increases consumer surlus. (ii) If < (n min ), the rice ceiling binds regardless of the realized demand and our corresonding analysis in section 4 carries over unchanged. It follows that, with demand uncertainty, if (n min ; ) < (n max ) and n min > c(b C)=(b + 2c), in the region where < (n min ); the rice ceiling that maximizes exected consumer surlus is given by (5). But, if n min < c(b C)=(b + 2c), an otimal rice ceiling cannot be strictly lower than (n min ) as exected consumer surlus is increasing in in this case. (iii) Let us now focus on 2 [ (n min ); (n max )]. As (n min ; ) 2 [ (n min ); (n max )], we can identify two sub-regions as otential locations for the otimal rice ceiling, [ (n min ); (n min ; )] and ( (n min ; ); (n max )]. a) Consider rst 2 [ (n min ); (n min ; )]. The rice ceiling binds regardless of the value of. However, deending on, the intervention may lead to excess demand or excess suly. More recisely, for a given rice ceiling, there exists a seci c value of = n () for which () =. This is the same n () as the one resented in (6). If () or, equivalently, if n (), then there is excess suly, so that consumer surlus is determined by the quantity demanded, q d (). If < () or, equivalently, if > n (), then there is excess demand and consumer surlus is determined by the quantity sulied, q s (; ). Denote by CS L dcs L d the exected consumer surlus in this case. Then, = 1 b (B ) b + c bc L d (n (b + c) [c(b C) (b + c) ( C)] ()) + (1 F(n ())), bc 2 where L d (n ()) is de ned in section 4. b) For 2 ( (n min ; ); (n max )] the rice ceiling may bind or not deending on the value of. More recisely, there exists a cut-o value of, let it be n (; ), imlicitly de ned by (n ; ) = so that for < n (; ), the rice ceiling does not bind and the quantity traded is q d ( (; )). For n (; ), the intervention may lead to excess demand or excess suly deending again on the value of. Seci cally, there exists another value of = n () for which () =. As in art a) above, this is the same as the one resented in (6), and n (; ) < n (). If n (; ) n (), then there is excess suly so that consumer surlus is determined by the quantity demanded, q d (). If > n (), then there is excess demand and consumer surlus is determined by the quantity sulied, q s (; ). Let CS H (11) be 17

18 the exected consumer surlus in this case. Then, dcs H d = 1 b (B )(F(n ()) F(n b + c (; ))) bc L d (n ()) + 1 b L d (n (; )) (12) c(b ) (b + c)( C) + (1 F(n ())) ; c 2 where n (; ) = [(b + c) cb bc]=c and L d (n ()) = R n () n min df(). In the online aendix we show that for n min > c(b C)=(b + 2c), the globally otimal rice ceiling is strictly lower than (n min ) and given by (5), whereas for n min C)=(b +2c) it lies in the interval [ (n min ); (n max )] and, deending on the arameter values, it is given by either dcs H=d = 0 or dcsl =d = 0.21,22 Let us now brie y turn to the case where (n min ; ) (n max ). As before, there are three sub-regions of rices where the otimal ceiling could lie. The analyses for > (n max ) and < (n min ) echo cases (i) and (ii) above and the corresonding results carry over unchanged. For 2 [ (n min ); (n max )], as (n min ; ) > (n max ), the rice ceiling binds regardless of the realization of. The intervention may lead to excess demand or excess suly deending on the value of. Intuitively, the analysis in art iiia) above alies now to the entire range. It is straightforward that the exected consumer surlus in this case is given by CS L. We are now ready to state the following result. Proosition 3 With demand uncertainty only, the otimal ro-consumer rice ceiling with imerfect cometition is weakly increasing in the degree of cometition. With no uncertainty, the otimal ceiling is indeendent of. But, in the resence of uncertainty, the otimal rice ceiling is weakly increasing in the degree of cometition and, therefore, weakly lower than the one in the erfectly cometitive market. c(b A decrease in the otimal ceiling generates signi cant gains in consumer surlus under high demand realizations. If the otimal ceiling is lower than (n min ; ), so that it binds regardless of the realized demand, then it is indeendent of the degree of cometition and is weakly lower than in the cometitive case. If the otimal ceiling is higher than (n min ; ), it may bind or not deending on the realization of demand and it is smoothly increasing in the degree of cometition. Suose that the rice ceiling that is otimal under erfect cometition were set in an imerfectly market. For demand realizations for which this ceiling does not bind under erfect 21 In the secial case where (n min ; ) is the otimal ceiling, both dcs H=d = 0 and dcsl =d = 0 hold. 22 In section 4, we found that, with erfect cometition and demand uncertainty only, exected consumer surlus is single eaked. This single-eakedness result is robust with imerfect cometition. We show this in the aendix by roving that the sloe of the exected consumer surlus with imerfect cometition is weakly smaller than that with erfect cometition. 18

19 cometition, the rice that obtains under imerfect cometition 23 is higher than the erfectly cometitive market-clearing rice. It is therefore above the consumer-surlus maximizing rice (see section 3), so that lowering the ceiling bene ts consumers. This argument underlies the conclusion that the otimal ceiling is ositively related to the extent of cometition. In our imerfectly cometitive market model, the aggregate outut is a fraction of the cometitive suly. This result builds on the suly function cometition microfoundation and, in the aendix, we show that 2c=(c + c(2b + c)) for n 2. However, the reduced form model with an outut restriction can also cature a monooly market when = c=(b + c). 24 Therefore, the analysis in this section, alies straightforwardly to a market with a monooly sulier. 6 Extensions of the Benchmark Model 6.1 Ine cient Rationing This section exlores the imlications of ine cient rationing using our benchmark model. More recisely, we examine ro-consumer rice ceilings in a erfectly cometitive market with uncertain demand and deterministic suly allowing for ine cient rationing. As the analysis builds on section 4, we focus on di erences stemming from rationing ine ciency and we highlight its interaction with the extent of demand uncertainty in the determination of the otimal rice ceiling. 25 Provided rationing e ciency is great enough, the results from the benchmark case still hold qualitatively, although the level of the otimal ceiling may deend on the value of. However, we in down a cut-o rationing e ciency below which it is otimal to set a rice ceiling that never binds. We use the three candidate regions for the otimal rice ceiling seci ed in section 4. Since a rice ceiling in the high region, > (n max ), never binds, it can be seen immediately that exected consumer surlus for a rice ceiling in this region is una ected by rationing ine ciency and so, as in section 4, exected consumer surlus is given by (7). However, in the middle region suly may be a binding constraint and in the low region it surely is, so for these regions ine ciency of rationing lays an imortant role. When, for a given, suly binds, outut q s () = ( C)=c is rationed. If this suly 23 This is either the imerfectly cometitive equilibrium rice or the ceiling, deending on the value of. 24 This value of can be inned down by solving for the monooly outcome but it also obtains in the suly function microfoundation if we allow n! Full details of the derivations are relegated to the online aendix. The arallel case with deterministic demand but uncertain suly is also analyzed there, allowing for ine cient rationing. 19

20 is allocated to the consumers whose valuations of the roduct are highest, consumer surlus is given by CS L d in (4). Alternatively, however, suose qs () is urchased by the consumers with the lowest valuations amongst those willing to ay at least. Then the q s () = ( C)=c units available are urchased by consumers along the lowest art of the demand curve at and above, i.e., from q d () q s () to q d (). At q d () q s () consumer surlus er unit is B+ b[q d () q s ()], while at q d () it is zero. Taking the mean of B+ b[q d () q s ()] and zero, and multilying by q s (), consumer surlus is then b( therefore write the consumer surlus for any at which suly binds as CS L d + (1 )CS L d0 = C) 2 =2c 2 CSd0 L. We can ( C)[2c(B ) + (1 2)b( C)] 2c 2 CS L d () : (13) In the low region (where < (n min )) suly always binds and so CSd L () is the exected consumer surlus. If 6= b=2(b + c); the rice ceiling that solves the f.o.c. of the otimization roblem in this region is ^() = c(b + C) (1 2)bC 2(b + c) b : (14) Note that, for = 1, ^() reduces to the value seci ed in (5). To be a well-de ned local maximum, ^() must lie in the interval (C; (n min )) and exected consumer surlus must be concave at this rice. However, if < b=2(b + c); ^() < C and gives a CS L d ()-minimum. CS L d region. () is increasing for all 2 (C; (n min )), and so an otimal ceiling cannot lie in this Generalizing De nition 1, we introduce the following notation. De nition 2 Let n 0 () = [(b + c) b](b C) 2(b + c) b We can now establish the conditions under which ^() is a local otimum. With > b=2(b+c), if n min n 0 (), CS L d () is increasing for all < (n min ) and the critical value ^() is weakly larger than (n min ), which is inconsistent with the region (C; (n min )): Similarly, if 2 (b=2(b + c); b=(b + c)) then, since n min < 0, obviously n min < n 0 (), and so the critical value exceeds (n min ). However, if n min > n 0 () and > b=(b + c), as d 2 CS L d ()=d2 < 0, (14) is a well-de ned local maximum within the region (C; (n min )). This generalizes Lemma 2. In the middle region 2 [ (n min ); (n max )]. Reeating the derivation in section 4, but using (13) to allow for rationing ine ciency, we obtain dcs d d = c L d (n ()). [2(b + c) b] ( C) c(b C) c 2 (1 F(n ()). (15) 20

21 Using the above analysis we can now state the results for > b=(b + c). These ndings for high e ciency of rationing are consistent with those in section 4. Proosition 4 With erfect cometition and demand uncertainty only, suose rationing e ciency > b=(b + c). If n min > n 0 (), the unique rice ceiling that globally maximizes exected consumer surlus is given by (14) and always binds. If n min n 0 (), it lies in the interval [ (n min ); (n max )) and this otimal T e as (b + c) L d ( e) [(b + c) b] (B C)(1 F(0)) T 0. This roosition highlights the combined e ects of the e ciency of rationing and the degree of uncertainty on the otimal rice ceiling. The analysis ins down a cut-o level of rationing e ciency = b=(b + c) above which the same qualitative conclusions aly as in the benchmark case. 26 is relatively small (n min Provided rationing is more e cient than this level, if uncertainty > n 0 ()) a rice ceiling in the low region, that is sure to bind, is otimal. A situation like this is illustrated in Figure 1B. Alternatively, if uncertainty is greater (n min n 0 ()), the otimal ceiling is in the middle region and may or may not bind. The critical level of uncertainty, as reresented by n 0 (), is decreasing in. Thus, the greater is rationing e ciency, the greater will be the maximum amount of uncertainty for which the otimal ceiling will lie in the low region, and bind for sure. We now consider the otimal rice ceiling when b=(b + c). Then, from (15), dcs d ( ((n max )))=d = 0 and d 2 CS d ( ((n max )))=d 2 < 0, so that ((n max )) is a local maximum on 2 [ (n min ); (n max )]. Also, from (13), dcs d ()=d > 0 for all < (n min ). Finally, if a rice ceiling is in the range > (n max ) it surely does not bind, and so the value of is not relevant. Therefore Lemma 3 still alies when b=(b+c), and it follows that a rice ceiling (n max ) is outcome-equivalent to any ceiling > (n max ). Hence, ((n max )) is a well-de ned maximum in the region [ (n min ); (n max )], and any rice ceiling higher than (n max ) yields the same outcome, whereas CS d is lower for < (n max ). This roves the next result. Proosition 5 With erfect cometition and demand uncertainty, if b=(b+c), any rice ceiling (n max ) globally maximizes exected consumer surlus. This result holds regardless of the amount of demand uncertainty. If rationing e ciency is low enough, rice regulation has a large allocative cost which reduces the scoe for roconsumer intervention. In e ect, the regulator refers not to intervene in the market and the otimal rice ceiling is high enough so that it never binds. If b > c; this non-intervention 26 Note, that the requirement that > b=(b + c) is consistent with a wide range of ine ciency. If, for examle, b < c, demand being less stee than suly, it is consistent with random rationing ( = 1=2). 21