Subjective Price Search and Public Intervention ∗ Alberto Galasso The London School of Economics and Political Science February 10, 2003

Abstract We analyse the effects of an exogenous reduction in search costs given by a transfer from government to consumers in the spatial model presented by Gabszewicz and Garella (1986). We show that it exists an optimal level of public intervention and that transfers which are too large destroy the equilibrium. Secondly, we present a model where a firm is inside a residential area and another is outside. We examine how the presence of the outsider (about which consumers have imperfect information) affects the equilibrium price. We show that in this new setting we do not have problems of existence any more.

1

Introduction

In traditional competitive theory it is assumed that consumers have perfect information about the prices and the characteristics of the products sold in the market. Before buying a product the consumer knows precisely the prices charged by all the firms and therefore he is able to recognize the best offer without incurring any kind of cost. Moreover products are supposed to be identical, which means that consumers are not able to distinguish among them. Different ways have been proposed in order to remove these two assumptions and to analyze product diversity in markets where there is imperfect information. In these models the presence of imperfect and costly information gives firms market power that not only relaxes price competition but also incentives to differentiate the products. Gabszewicz and Garella (1986) present a reformulation of the classical Hotelling’s location model removing the assumption that buyers have perfect and costless information. Introducing elements of imperfect information into Hotelling’s Main Street, different results are obtained with respect to the standard model. The search process proposed is based on the assumption that consumers have full information on their nearest supplier’s price while the cost of finding the ∗ I am grateful to Jean Gabszewicz, Clara Graziano and Enrico Sette for helpful comments and discussions.

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price of an alternative supplier is positive and linearly related to the distance between consumer and supplier. The analysis of Gabszewicz and Garella focuses on two different information endowments to the consumers. In the first one they know with certainty the range within which prices may fall, in the second they center the range on the first price observation. From these assumptions Gabszewicz and Garella show that in the first case the only possible equilibrium is the price at which all consumers prefer to buy at their local supplier rather than to search. The existence of such an equilibrium has to meet a condition similar to the one in d’Aspremont, Gabszewicz and Thisse (1979) for the classical Hotelling’s spatial model: firms have to lie sufficiently far apart. For the setting where consumer center their subjective probability distribution of the unknown price on the one observed at the local shop, a price equilibrium never exists whatever the firms’s locations. As Gabszewicz and Garella (1986) adapted the model of Hotelling (1929) to imperfect information, Wolinsky (1983) introduced uncertainty in the fixed price circular product space model. In this setting, in contrast with the linear one, the existence of a symmetric price equilibrium (at which the available brands are spaced equally around the circle) is always guaranteed. In a subsequent paper Wolinsky (1986) presented another model where consumers have to bear a search cost to find out about prices and characteristics of the various products. Here the symmetric equilibrium exists only if the incentive to search that consumers have is not too little. An important feature of this model is that, as the number of firms gets larger and the search cost that consumers have to sustain becomes negligible, the equilibrium price tends to the competitive one. The importance of this result is that it does not present the paradoxical conclusion (monopoly price) which was proposed by Diamond (1971). An interesting extension to Wolinsky (1986) is presented by Anderson and Renault (2000) introducing a fraction of fully informed consumers in the model. The price equilibrium, in this new framework, is negatively affected by the size of the atom of informed consumers: the larger the atom, the higher the price. This result is in stark opposition with respect to part of the literature (see for instance Wolinsky (1983) and Stahl (1989)) but it is also supported by other contributions (as Gabszewicz and Grilo (1992) or Fudenberg and Tirole (1984)). A synthesis of the Hotelling’s spatial competition and the Chamberlinian monopolistic competition in markets where consumers are not perfectly informed is proposed by Anderson and Renault (1999) in the Bertrand-Chamberlin-Diamond model. They demonstrate the existence of a symmetric price equilibrium which is an increasing function of the search costs and a decreasing function of the number of firms in the market. Moreover, they study the relation between an increase in product diversity and the value of the equilibrium price. The result is a clear U-shape of the symmetric equilibrium price for increasing level of product variety. The paper is organized as follows. In Section 2 we present the model and the principal results of Gabszewicz and Garella (1986). In Section 3 and 4 we propose two extensions to the model . Firstly we introduce an exogenous reduction in search costs given by a transfer from government to consumers. 2

We will show that it exists an optimal level of public intervention and that transfers which are too large destroy the equilibrium. Secondly we present a model, following an idea in Gabszewicz and Thisse (1986), where a firm is inside a residential area and another is outside. We will show how the presence of the outsider (about which consumers have imperfect information) affects the equilibrium price and that the equilibrium remains for any level of transfer.

2

The model

Consider a line of length L and two sellers, 1 and 2, located at respective distances a and b from the endpoints of the line. The two sellers do not have production costs and they sell the same homogeneous product. Consumers are evenly distributed along the line and each of them is indexed by t ∈ [0, L]. Each consumer consumes exactly a single unit of this commodity, irrespective of its price. The transportation costs which characterize the Hotelling’s model are here substituted by search costs that consumers have to sustain. Indeed, the single consumer learns at no cost the price quoted by the seller nearest to him (the local seller ) and he can get the information about the price of the most distant one at a cost which varies linearly with the distance from him. Any buyer can either buy at the known price from the local seller or sustain the search cost in order to know the price set by the most distant and then decide where to buy. This decision depends on the expectation of finding a higher or a lower price there, as well as on the search cost. This consumers’ information pattern generates a partition in the market: buyers located at the left of the point L+a−b know a no cost the price quoted by 2 seller 1; those located at the right know at no cost the price of seller 2. The mid point L+a−b , which lies at the center of the distance between the two shops, 2 is called informational frontier and it separates the natural markets of the two firms. It is worth noticing that if search costs were infinite the two sellers would operate like monopolists on their natural market while, in the opposite case of zero search costs all buyers would buy from the low price firm. The latter case leads to the results of the Bertrand (1883) model where consumers are perfectly informed about prices and where the two firms charge a price equal to their marginal cost. h h i i (L+a−b) Defining A1 = 0, (L+a−b) = , L , we indicate with d(t) and A 2 2 2 the distance between the farthest shop and consumer t’s location. Precisely we have that d(t) = L − b − t if t ∈ A1 and d(t) = t − a if t ∈ A2 . The search cost is assumed to be a continuous monotone increasing function of the distance and it is denoted by c(d(t)) or, to shorten the notation, by c(t). Once described the model, the analysis of Gabszewicz and Garella (1986) proceeds considering two different cases: in the first one consumers are supposed to know the range within which prices may fall; in the second case they have

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not this information and they center their expectations about the price range on their first price observation.

2.1

Information about the Price Range

In this information endowment it is assumed that the customers’ expectations about the unknown price are derived from a subjective probability function F (p) defined over a range [0, p] with a density f (p) > 0 over the same range. In this context we can define the expected gain from search to customer t ∈ A1 as: φ(pe1 , t) = pe1 −

Z

0

p f1

f (p2 ) · [p2 + c(d(t))] dp2 −

Z

p

p1 f

f (p2 )dp2 · [pe1 + c(d(t))] ,

where pe1 stands for the observed value of p1 at location 1, and p2 is the unknown price at shop 2. The definition of φ(pe2 , t) is immediate by analogy1 . The consumer t ∈ A1 will search if and only if φ(pe1 , t) ≥ 0. The reservation price, denoted by p∗1 (t) for t in A1 and by p∗2 (t) for t in A2 , is defined as the price at which consumer t is exactly indifferent between buying without search at the local shop and searching. This value can be computed setting φ(pe1 , t) = 0. It can be easily checked that both p∗1 (t) and p∗2 (t) are continuous, monotone and decreasing functions in t and they reach their minimum values at t = L+a−b . 2 Let us define the reservation in the middle point of ¡ ¢ price∗ of ¡ the consumer ¢ the segment as p∗ = p∗1 L+a−b = p2 L+a−b . If both sellers were to quote 2 2 p∗ or a price below it, then no customer would engage in search, and each seller would serve his own natural market. On the contrary, if a seller sets a price above p∗ , then a portion of the customers population from his natural market would search and acquire price information. This¤portion is £ thereby complete ¤ £ given by the segment tb1 (p1 ), L+a−b , tb2 (p2 ) for seller for seller 1 and L+a−b 2 2 2, where tb1 (p1 ) denotes that buyer which is indifferent between searching and buying at shop 1 at the known price p1 . The values of tb1 (p1 ) and tb2 (p2 ) are simply the solutions of the equations p∗1 (t) = p1 and p∗2 (t) = p2 . The customers with the highest incentive to search are those located around the middle point L+a−b . Accordingly if p1 > p∗ and p2 > p∗ then the buyers 2 located between tb1 (p1 ) and tb2 (p2 ) all undertake search and get full information. In this case they all are going to buy from the shop quoting the lowest price, while the uninformed customers are going to buy from their local supplier. As we can notice in this model, for the informed group of the consumers the firms incur a Bertrand price competition while, for the uninformed part of them, firms keep 1 Both

expressions may be also written as: R pi )] − 0pei pj f (pj )dpj . φ(e pi , t) = pei − c(t) − pei [1 − F (e

This expression distinguishes clearly the various components of the expected utility: the price observed, the search cost and the probability to observe a lower (or a higher) price.

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a monopoly power. In fact, the demand addressed to seller 1 is easily inferred to be equal to D1 (p1 , p2 ) = L+a−b if p1 and p2 ≤ p∗ or p1 = p2 ; equal to tb2 (p2 ) if 2 p∗2 (L) > p2 > p1 and p2 > p∗ ; equal to tb1 (p1 ) if p∗1 (0) > p1 > p2 and p1 > p∗ and, finally, equal to zero if p1 ≥ p1 (0). By symmetry D2 (p1 , p2 ) = L − D1 (p1 , p2 ). Analyzing these demand functions, Gabszewicz and Garella (1986) show that, whenever it exists, a non cooperative price equilibrium is uniquely determined by µ µ ¶ ¶ L+a−b L+a−b p∗ = p∗1 = p∗2 . 2 2 Let c(t) = c [L − b − t] if t ∈ A1 and c(t) = c(t − a) if t ∈ A2 , where c > 0. Given the range [0, p] within which the prices are known to fall, setting p1 = 1, and assuming that the subjective probability distribution takes the form of the uniform distribution over [0, 1], it is demonstrated that the equilibrium is given by the pair of prices: ³p ´ p (p∗ , p∗ ) = c (L − a − b), c (L − a − b) .

It is also checked that the existence of this equilibrium is subordinated to the following two conditions: b + 3a ≤ L and a + 3b ≤ L. Considering a simple example where a = b and L = 1, the required condition is just to have a ≤ 1/4. So if the firms are too close to each other then a price equilibrium does not exist, exactly as in the standard Hotelling’s model. If firms are allowed to choose their locations before choosing prices, both of them can anticipate the consequences of their locational choice on the equilibrium prices. Considering the equilibrium pricesp (p∗1 , p∗2 ) in the revenue function 1 we have that R1 (a1 ) = p1 D1 (p1 , p2 ) = L+a21 −b c (L − a1 − b), so that dR da ≥ p L+a−b2 c (L − a − b2 ), so 0 ⇔ 3a + b ≤ L. Similarly R2 (b2 ) = p2 D2 (p1 , p2 ) = 2 1 that dR ≥ 0 ⇔ 3b + a ≤ L. db Consequently, if the effect of the location choice on price competition are taken into account, both sellers have an incentive to cluster towards the competitor, at least in the whole domain where price equilibria exist. But if the firms are too close to each other, as in the Hotelling’s model, a price equilibrium does not exist. This result is not surprising since it is possible to reach the same conclusions for the Hotelling spatial model with perfect information [see d’Aspremont, Gabszewicz and Thisse (1979)].

2.2

Imperfect Knowledge of the Price Range

In their paper Gabszewicz and Garella (1986) consider also the case where consumers, after having observed the prices at their local shops, form their expec5

tations about the unknown prices by relating those expectations to the observed prices. More precisely it is assumed that the probability distribution F (p) is uniform with its center at the price observed at the local shop. This assumption corresponds to a situation in which the buyers do not know the price range and infer it from the first price observation. This idea has been formalized letting the support of the uniform distribution F (p) be [0, 2pe1 ]. In this framework Gabszewicz and Garella (1986) demonstrate that a price equilibrium never exists, whatever the firms’ locations. This result reflects the fact that firms can manipulate the buyers’ search process in a more elaborate way. In fact firms can affect the buyers’ beliefs about the prices of the rival firm and gain a larger profit by setting a price higher than p∗ . In both Hotelling’s model (where consumers are perfectly informed) and the present one, there is a wide domain of location parameters entailing the absence of a price equilibrium. However, the mechanism causing the existence of a price cycle is very different in the two cases. In Hotelling’s case, it relies on either firm’s incentive to set a price lower than the one of its own competitor when the seller are too close to each other. In Gabszewicz and Garella (1986), whatever the locations, the equilibrium obtains at that price at which no consumer searches. But when the firms are too close to each other, that equilibrium is destroyed as either seller has an incentive to quote an higher price, although losing some customers. Thus the winning strategy is not to undercut prices but, on the contrary, to set a price which is higher than the candidate for an equilibrium. The intuition behind this result is that if consumers center their expectations at the price observed at the local shop then an increase in the price set by a firm increases the expected price estimated by its local consumers reducing in this case the expected gains from search. Another relevant difference between Hotelling’s model and Gabszewicz and Garella’s one is that while in Hotelling (1929) the equilibrium prices differ at the two shops for all asymmetric locations, in Gabszewicz and Garella (1986) prices are equal in both shops whether or not they are symmetrically located.

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Reducing Search Costs

The first extension we are going to propose to the model of Gabszewicz and Garella (1986) consists in a refinement of the search cost function. The idea is to model an exogenous reduction in search costs given by a transfer from government to consumers. More precisely, since the presence of imperfect information leads to results which differ from standard Bertrand equilibria, the question we would like to answer is the following one: is it possible to reduce distortions by means of a public intervention? Trivially, if each subject was to receive a transfer exactly equal to his search cost, information would be no longer costly and we would come back to the general Bertrand setting with marginal cost pricing. Instead, the situation we are going to depict is the one where public sector is not able to discriminate across consumers and therefore transfers do not differ across them.

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3.1

The model

We use the first model proposed in Gabszewicz and Garella (1986): customers are supposed to know with certainty the range, [0, p], within which prices fall. Moreover (as in the original paper) we set p = 1 and we assume that the subjective probability distribution, F (p), takes the form of the uniform distribution over [0, 1]. We introduce a public transfer of amount equal to K received by each consumer. This leads to new search cost functions: c(t) = c(L − b − t) − K = c(t − a) − K

if t ∈ A1 if t ∈ A2 .

(1)

Setting K = 0 we return to the original model of Gabszewicz and Garella (1986). The reservation prices p∗1 (t) and p∗2 (t) can be obtained by equating φ(pe1 , t) and φ(pe2 , t) to zero. It is found that: p p∗1 (t) = 2 [c (L − b − t) − K] (2) and

p∗2 (t) =

p 2 [c (t − a) − K].

(3)

The lowest reservation price in the consumers population is easily found to be p∗

¡ L+a−b ¢ 2

=

q £ ¡ ¢ ¤ q ¡ ¢ 2 c 2L−2b−L−a+b − K = 2c L−a−b − 2K 2 2 =

p c (L − a − b) − 2K.

(4)

Using formulas (2) and (3) we can easily identify the buyers who are indifferent between buying without search at the local shop and searching. Indeed,

and

p21 = 2c(L − b − t) − 2K =⇒ b t1 = L − b − p2 K b t2 = 2 + + a. 2c c 7

p21 K − 2c c

(5)

(6)

Figure 1: Introducing K we shift downward the curve of the reservation prices reducing the value of p∗ and increasing the number of informed consumers. It is worth noticing that an increase in K implies an enlarged value for b t2 and a reduced b t1 . Therefore, public intervention makes the fraction of informed consumers get larger. This reduces firm market power and renders competition keener. We compute pb1 , the price which maximizes the revenue of Firm 1 given that Firm 2 charges p∗ , as the solution to the problem: ³ ´ p2 p2 p2 max p1 L − b − 2c1 − Kc =⇒ L−b− 2c1 − Kc − c1 = 0 ⇒ 2cL−2cb−2K = 3p21 p1

obtaining pb1 =

r

2 [c (L − b) − K]. 3

(7)

Using the same reasoning of Gabszewicz and Garella (1986), we claim that the pair of prices (p∗ , p∗ ) is a price equilibrium if and only if p∗ ≥ pb1 , i.e., iff p p = c(L − a − b) − 2K ≥ ∗

or

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r

2 c [(L − b) − K] = pb1 3

K ≥ 3a + b. c By an analogous reasoning we require that r p 2 ∗ (cL − ca − K) = pb2 p = c(L − a − b) − 2K ≥ 3 which implies L−4

(8)

K ≥ 3b + a. (9) c Having computed these two conditions we conclude with the following: ³p ´ p c(L − a − b) − 2K, c(L − a − b) − 2K Proposition 1 The pair of prices (p∗ , p∗ ) = is a price equilibrium if and only if (8) and (9) simultaneously hold. L−4

3.2

Economic intuition and policy

The main idea behind the results we have provided is that a reduction in information costs (through public intervention) diminishes the equilibrium prices only if it does not exceed a specific threshold value. In other words, we argue that an excessive value of K may be a source of instability . To better describe this concept we have computed a numerical example where the two firms are equally spaced (a = b = 18 ) and L and c are normalized to 1 (L = c = 1). K p∗ 0 0.86 1/10 0.74 1/8 0.7 1/6 0.64 TABLE 1: Effect of Public Intervention

pb 0.76 0.71 0.7 0.68

We can observe from Table 1 that as K increases, the equilibrium price, p∗ , decreases and the price strategy which maximizes the revenues, pb, gets smaller as well. It is worth noticing that K = 18 (value that can be computed in this symmetric setting equalizing (4) and (7)) is the threshold value to have stability. Higher values of public intervention imply price cycling because, for K > 18 , (p∗ , p∗ ) is no more a Nash equilibrium. The more expensive is the search (parameter c) the larger is this threshold value. Moreover, it increases as L gets larger and either a or b diminish. This means that public intervention will be more effective the higher are the search costs borne by consumers. On the contrary, in markets where these costs are not so relevant, introducing K is more likely to be a source of instability.

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To conclude our analysis we want now to answer to a particular question: is it possible to reduce the equilibrium price to zero? More precisely, we would like to find some conditions on the parameters such that public intervention completely removes imperfect information in the product market and restores the standard Bertrand equilibrium. If K is so large that for all consumers the search cost becomes zero then (trivially) marginal cost pricing holds again. But if we consider a government budget constraint2 such that there exists always a fraction of consumers not completely informed, then p∗ is never zero. Proposition 2 For each value of c, a, L and b, it does not exist a value of K such that p∗ goes to zero. Proof. First of all we recall that, in order to have a Nash equilibrium, transfers have to satisfy both (8) and (9), which we rewrite as: K≤

c(L − b − 3a) 4

(8’)

K≤

c(L − a − 3b) . 4

(9’)

Moreover we know that p∗ = 0 ⇔ which implies

p c(L − a − b) − 2K = 0

c(L − a − b) . (10) 2 To have p∗ = 0 as an equilibrium, K has to satisfy the three conditions (8’), (9’) and (10). But a value of K satisfying all these conditions does not exists! Indeed, if (8’) holds then we have that K=

c(L − 3a − b) c(L − a − b) ≥ ⇔b−a≥L 4 2 and if (9’) holds, then

(11)

c(L − 3b − a) c(L − a − b) ≥ ⇔ a − b ≥ L. (12) 4 2 But since L ≥ a + b ≥ 0, (11) and (12) cannot hold. Impossibility follows. Our main conclusion is that, in this particular setting, a government intervention has not always a positive effect on consumers. If the value of K is too large, its effect is to destroy price stability. Moreover, cases where equal public 2 For

instance, K < max {c (L − b) , c (L − a)}.

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transfers (considering the budget constraint of note 1) completely remove the effects of imperfect information do not exist. An interesting remark can be done considering a consumer association. High levels of K can be interpreted as a public institution of such an association. Indeed, with a transfer large enough (larger than the value of p∗ that we observe without intervention), government reduces to zero search costs for a fraction of consumers. This implies the introduction of an atom of perfectly informed consumers around the point L+a−b . But, as we have argued with 2 Proposition 1 and Proposition 2, as long as there remain consumers for which information is costly, there will be price cycling. Therefore, we can conclude that the institution of a consumer association (by means of a large public transfer) is a source of instability in the model presented3 .

3 It is necessary to stress again that if the transfer is very large, we reduce to zero the search cost for each consumer. This corresponds to a consumer association covering the entire market. The equilibrium price in this case will be (trivially) zero.

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4

An Insider-Outsider Model

The second extension we are going to present considers the existence of a firm located outside the segment [0, L] . A similar case with perfect information is presented in Gabszewicz and Thisse (1986). The intuition is the following one: consider a road along a closed valley and a population spread uniformly along that road between point zero and point L. Two shops compete selling a homogeneous product. Firm 1 is established inside the residential area, consumers know the price it charges without incurring any cost. Firm 2 is not allowed inside the residential area because of zoning regulation. Its price is not common knowledge across consumers: they have to incur a cost, c(L − t + b), in order to observe it. It is clear that in this model we have to consider a unique reservation price: p∗ (t) (as represented in Figure 2). Indeed, p∗ (t) stands for the price rendering consumer t exactly indifferent between buying without search at the local shop and searching. For each price, pb, set by Firm 1 the optimal response of Firm 2 is to charge a price slightly lower than that: p2 = pb − ε. Indeed, if it charges a price larger than pb , the entire market is served by Firm 1; if it charges the same price it sells to half of the informed consumers and if p2 is lower than pb it sells to the entire fraction of informed consumers4 . Therefore, for each price charged by Firm 1, once we have computed the indifferent consumer we obtain the two market shares of the two competitors: Firm 1 serves the uniformed consumers while Firm 2 serves the informed ones. Using the usual technique we compute the reservation price as p p∗ (t) = 2c(L + b − t).

(13)

It is worth noting that the location of Firm 1 does not affect p∗ (t). Moreover, p (t) gets larger as b increases, that is, as Firm 2 is located farther apart from L. From (13), given p1 (the price charged by Firm 1), we obtain the corresponding indifferent consumer as ∗

t(p1 ) = L + b −

p21 2c

= L

√ 2cb √ if p1 ≤ 2cb

if p1 >

(14)

and we compute the price that maximize the revenues for Firm 1 as: 4 Notice

that the pair (b p, pb − ε) is not really an equilibrium. The reason is because ε is not properly defined. As in the case of a Bertrand duopoly where the two firms have different constant marginal costs the optimal strategy for the firm with low costs is to charge the marginal cost of the other minus a small epsilon. But for each epsilon it is possible to find another (smaller) epsilon such that for the firm is better to deviate!!

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Figure 2: Firm 2 is located outside the segment at a distance from the endpoint L equal to b.

max pt(p) p

whose solution is r

2 L c(L + b) if b ≤ 3 2 √ L 2bc if b > . = 2

pb =

(15)

To compute pb we have considered the fact that all the consumers will buy from Firm 1 if t(b p) ≥ L and this happens if and only if

L 2 (L + b) ≥ L ⇔ b ≥ . (16) 3 2 The result we have obtained is that the presence of an outside alternative affects the demand addressed to the local firm only if the outsider is located close enough to the endpoint. Conversely, the presence of the outsider always affects the revenues of the local seller through a price which is lower than the one of monopoly. Indeed, the price charged by Firm 1 tends to infinite (monopolistic price) only when b goes to infinite. These results are supported by economic intuition: the farther is the outsider, the bigger is the market power of the local 13

seller. In fact, as b gets larger profits of the local seller increase: initially because of both a larger demand and a higher price then, when market is saturated, only because of the price effect. Consider now the role of public sector in this alternative setting. Reinstating equal public transfers, as we have done in the previous model, we have that: p p∗ = 2 [c (L + b − t) − K] and

K p2 − . c 2c This implies that the price charged by Firm 1 will be t(p) = L + b −

r

L K 2 [c (L + b) − K] if b ≤ + 3 2 c r L K 2 (cL − K) if b > + . = 3 2 c

pb =

(17)

A relevant difference between this model and the previous one is that in this new setting we have no longer problems of instability. As K increases the price charged by Firm 1 gets smaller and, to the limit, it approaches to zero without displaying any cycling phenomena. Moreover, if the public transfer is larger than cb then, as before, we can interpret it as a public intervention to establish a consumer association. Indeed, an atom of informed consumers appears near the endpoint L. It is worth noting how, in this new context, the introduction of such an association always reduces the equilibrium price5 .

5

Conclusion

Gabszewicz and Garella (1986) adapted Hotelling’s location model introducing imperfect information. In this paper we have analyzed the effects of a public intervention in the form of transfers from government to consumers. We have shown that in the original model of Gabszewicz and Garella there exists an optimal level of public transfers and that a transfer too large destroys price stability. Secondly we have presented a model where a firm is inside a residential area and another is outside. We have examined how the presence of the outsider (about which consumers have imperfect information) affects the equilibrium 5 Again we stress that with a transfer higher than c(L + b) all consumers have zero search costs and we come back to the standard Bertrand competition with marginal cost pricing.

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price. We have shown that in this new setting we do not have any more problems of existence and stability.

References [1] Anderson S.P., Renault R., 1999, Pricing, Product Diversity and Search Costs: a Bertrand-Chamberlin-Diamond model, RAND Journal of Economics, 4, 719-735. [2] Anderson S.P. and Renault R., 2000, Consumer Information and Firm Pricing: Negative Externalities from Improved Information, International Economic Review, 41, 721-742. [3] d’Aspremont, Gabszewicz and Thisse, 1979, On Hotelling’s Stability in Competition, Econometrica 47, 1145-1150. [4] Bertrand J.L.F, 1883, Théorie mathématique de la richesse sociale par Léon Warlas: Recherches sur les principes mathématiques de la théorie des richesse par Augustin Cournot, Journal des Savants 67, 499-508. [5] Diamond, P.A., 1971, A Model of Price Adjustment, Journal of Economic Theory, 3, 156-168. [6] Fudenberg, D. and Tirole, J., 1984, The Fat Cat Effect, the Puppy Dog Play and the Lean and Hungry Look , American Economic Review, 74, 361-366. [7] Gabszewicz J.J. and Garella P., 1986, Subjective price search and price competition, International Journal of Industrial Organization 4, 305-316. [8] Gabszewicz J.J. and Grilo I., 1992,Price Competition when Consumers are Uncertain about which Firms sells which Quality, Journal of Economics and Management Strategy, 4, 629-650. [9] Gabszewicz and Thisse,1986, On the Nature of Competition with Differentiated Products, The Economic Journal, 96, 160-172. [10] Hotelling H., 1929, Stability in Competition, Economic Journal 39, 41-57. [11] Stahl, D.O., 1989, Oligopolistic Pricing and Sequential Consumers Search, American Economic Review, 79,700-712. [12] Wolinsky A., 1983, Prices as Signal of Product Quality, Review of Economic Studies, 50, 647-658. [13] Wolinsky A., 1986, True monopolistic competition as a result of imperfect information, Quarterly Journal of Economics, 101, 493-511.

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