Supply location uncertainty and consumer search

Supply location uncertainty and consumer search

Barbara M. Winkel

A thesis submitted to the Faculty of Economics in partial fulfilment of the requirements

for the degree of

Master of Science

Graduate Programme in Economics University of Groningen

Abstract

In this thesis we discuss a Hotelling model with consumer side uncertainty about supply location. We consider a Hotelling line with two firms that are located at the extremes. There is one good, which is sold by only one of the firms. Some of the consumers are informed about which firm sells the good, whereas the rest only knows that one of the two firms sells the good, with equal probability. We analyse search behaviour by consumers, and price setting behaviour by the producer. Furthermore, we discuss the welfare implications of increasing market transparency, which is measured directly by the proportion of informed consumers, and indirectly by the transportation cost.

Acknowledgements

I would like to thank a few people, who, each in their own way, have been instrumental to the successful completion of my thesis. Firstly I would like to thank my supervisor, Bert Schoonbeek, for his guidance over the past three and a half years. He introduced me to the wonderful world of Industrial Organization, and made sure I would never want to leave it. His insights, suggestions, and faith in my ability were essential ingredients in the finished product.

Secondly I would like to thank SOM for providing me with the opportunity to broaden my horizons, first in the Honours programme, and later in the Research Master. They have also provided me with a space to work in, which has been a quiet haven for me during my work on my thesis.

Thirdly, I would like to thank my parents for all their financial and non-financial support over the years - surely I could not have done this well without them. I want to thank my father in particular for reading through parts of my thesis and providing me with a different perspective on some issues. His comments have certainly improved the legibility of my thesis.

Finally, my thanks go to my partner Richard, for all his love and support, and for taking the time to proofread sections of my thesis, even if these must surely have bored him horribly.

Table of Contents

Abstract i

Acknowledgements ii

Table of Contents iii

1 Introduction 1

2 Literature on spatial models and models of imperfect information 5

2.1 Spatial models . . . 5

2.2 Supply and demand uncertainty . . . 7

2.3 Transparency . . . 9

2.4 Search models . . . 12

3 A model of supply location uncertainty 16 3.1 The consumer problem . . . 17

3.2 The producer problem . . . 21

3.3 Comparative statics and welfare . . . 24

4 Conclusion 28 Bibliography 30 A Appendix 33 A.1 Proof of Lemma 1 . . . 33

A.2 Proof of Lemma 2 . . . 36

A.3 Proof of Proposition 1 . . . 40

1

Introduction

Imagine the following scenario: you are walking around when you see a billboard advertising a TV that you really like. You know of two stores that might stock this TV, each located at opposite sides of town. You know that only one of these stores will actually sell the good. You are currently somewhere in between these stores. Let’s also assume that the billboard states the price of the item you want, so that all you need to worry about is where to get the good. Now, what do you do? Do you go look for the good in hopes of finding it? And if so, will you visit both stores or will you give up after just one? Which store will you visit first? And, if you do find the good, will you still want it?

This is a common situation for many consumers: they know exactly what they want, but not where to find it. It is exactly this situation that we will examine now. It is important to note first that our model is interesting not only from the point of view of consumers and producers, but also for the government. The level of market transparency – which is the level to which consumers and producers are informed about important market characteristics such as price and product location – can have an effect on welfare. An increase in transparency can be both positive for welfare, for example if consumers become more informed about prices, which can make the market more competitive; or an increase in transparency can be negative for welfare, for example if producers become more informed about each other’s prices which may facilitate collusion and raise prices. So, it may be interesting for a government to try to change the level of market transparency, if it is expected that this will raise welfare. Stefanski, de Vries and van der Waarden (2002) provide a Dutch overview of policy issues regarding market transparency. This thesis can be seen as an addition to this discussion.

good, minus the costs incurred to get it.1 _{Schultz finds that as the proportion of informed}

consumers increases, consumer surplus increases and profits decrease. This decrease in profits is partially due to the fact that products become less differentiated as transparency increases.

Neither of these models allows consumers to search for the good after visiting the first store. Of course, in real life consumers can search. This is discussed in Burdett and Judd (1983), among others. In their model producers all sell identical goods. Consumers are uninformed about the prices set by producers, and may search for the lowest price at a positive search cost. The authors find that if consumers only visit one producer, producers will act like monopolists. After all, each producer is effectively a monopolist to every consumer that comes to his shop, because no consumer will go to any other producer. If consumers search at least twice, prices are as they would be in a perfectly competitive market. This implies that as consumers become more informed (by searching more often) prices fall. This is due mostly to the fact that the market becomes more competitive if consumers are more informed.

In contrast to the models discussed above where consumers are imperfectly informed about prices, we will consider a model in which consumers are informed about prices, but not about the exact location of the good. It is interesting to note that we assume the exact opposite from Schultz (2004): he assumes that consumers are uninformed about prices and product characteristics, but do know where to find the good; we assume that consumers know prices and product characteristics, but not where to buy the good. We consider the following situation: there is one good that all consumers want. There are two shops that might sell the good, but only one of these shops will actually hold the good in stock. This can be related to the TV example earlier: you want a certain brand of TV; there are two electronics shops that sell TVs; however, only one of these shops has a contract with the manufacturer to sell the particular brand that you want. We also assume that the shops are located at either end of a town that consists only of one long Main Street. Consumers are evenly spread out along this street.2 _{Some of these consumers will be}

uninformed about which shop sells the good, whereas others will be informed about this. The latter may have Internet access which makes them better informed, or maybe they visit the shops more regularly than uninformed consumers do. We assume that informed and uninformed consumers are evenly mixed. In order to buy the good, consumers incur a transportation cost, depending on how far away they live from the shop(s) they are visiting. This transportation cost can be seen as the cost of the petrol used to go to the shop, or a cost for the time spent shopping when the consumer could have been doing something else. Another interpretation for the transportation cost is that of a search cost: although all

1

So, if a consumer values the good at e10, and he has to spend e 8 to get it (this includes the price and possibly a transportation cost), then his surplus will be e 2. In this example the surplus is positive so the consumer will buy the good.

2

consumers will incur some transportation cost, uninformed consumers will probably incur more than informed consumers, because they are likely to have to travel further in order to buy the good.

There are several important questions to be asked with respect to this situation. When will informed consumers buy the good? When will uninformed consumers decide to search for the good? How many shops will they visit and in which order? Will consumers be better off if they are informed or if they are uninformed? From the producer side there are also several issues: How does the fact that some of the consumers are uninformed about supply location affect the price the producer sets and the profit he makes? If more consumers are informed, will profits be higher or lower? Will the economy as a whole be better off if more consumers are informed about supply location?

For ease of exposition, let’s call the shops shop A and shop B. Assume that shop B is the one that sells the good. We find that those consumers that are informed about the location of the good buy it if this gives them a positive surplus. This means that they will buy the good if the value they attach to it is higher than the cost of obtaining it (which is price plus transportation cost). Uninformed consumers will buy the good less often than informed consumers. One reason for this is that the expected benefit of visiting the first shop is lower for uninformed consumers: after all, they are not sure they will find the good at the first shop they visit. This means that uninformed consumers are less likely to visit a shop than are informed consumers. So, uninformed consumers will consume less. Also, uninformed consumers will visit the shop they are closest to first. If they do not find the good there they will move on to the other shop, if they derive a non-negative surplus from visiting this second shop. Uninformed consumers who are located closer to shop A than to shop B and who buy the good will incur a higher transportation cost than consumers located closer to shop B because they have to visit both shops in order to buy the good. This higher transportation cost means that uninformed consumers who live closest to shop A will buy the good less often than uninformed consumers who live closest to shop B.

We also find that the producer sets his price depending on how many consumers he wants to reach. This in turn depends on the consumers’ valuation of the good. For example, when the valuation is low, he is unlikely to attract any of the uninformed consumers who live closest to shop A unless he sets a very low price. Because he can make a higher profit by setting a higher price and attracting less consumers, he will decide to aim only at informed consumers and uninformed consumers living closest to him. When the valuation is very high the producer will set a price so that everyone will just buy the good (so if he set the price even slightly higher he would lose some of his customers). Also, for the most part price increases as the valuation for the good increases, but at a lower rate.3 _As

a result the number of consumers buying the good will increase as well.

Finally, we find that welfare increases as market transparency increases , i.e. as the proportion of informed consumers increases, or as the transportation cost decreases. As market transparency increases, consumer surplus (which for each consumer is the value of the good, if bought, minus the cost incurred to get it) increases. Profits and prices

3

also increase with an increase in market transparency. This means that if there were for example some government agency that could either increase the proportion of informed consumers (e.g. by giving everyone free Internet access) or decrease transportation cost (e.g. by subsidizing public transport), or both, then this would be beneficial to both the producer and consumers. This is in contrast to the result of Schultz (2004), who found that profits decreased as consumers became more informed. However, me must keep in mind that in Schultz producer locations are endogenous, and producers locate closer together as consumers become more informed. This has a negative effect on profits. Our findings also contrast to the findings of Burdett and Judd (1983) who found that prices decreased when consumers searched more; we find that as search becomes cheaper (if the transportation cost falls), prices become higher. This difference in findings may be easily explained by the fact that we consider the monopoly case, whereas Burdett and Judd consider a case with many firms. So, in their model, if consumers become more informed there will be more competition between firms, which causes prices to drop. In our model the monopolist simply extracts more surplus from the consumers if transportation costs fall.

2

Literature on spatial models and models of

imperfect information

We will now discuss some major economic theories that are relevant to the model we consider in the next session. First we will discuss spatial models; second, we will discuss models in which there is uncertainty. This can be uncertainty on the producer side about demand, or uncertainty on the consumer side about supply. Third we will discuss models of market transparency. Finally, we will discuss search models.

2.1

Spatial models

Spatial models find their origin in Hotelling (1929). Hotelling devised his model in a reaction to the work by Bertrand (1883), who stated that two competitors competing in price would keep undercutting each other’s prices until neither of them made a profit. This originated from the idea that if two goods were the same, consumers would buy from the producer setting the lowest price - so, each producer could capture the entire market by undercutting the other’s price just slightly. Hotelling did not believe this to be a realistic representation of the economy, arguing that, even if two products were exactly the same in themselves, they might still differ due to the location in which they were sold, or by the service offered by the producer, etc. As an extreme example, a certain car may be sold by a dealer in Europe, and a dealer in the United States. If a consumer lives in Europe he is likely to buy the car from the European dealer, even if the car is cheaper in the U.S. This is because, even though the car is the same no matter where it is sold, the location of the dealer is also a very important characteristic to consumers. So, a consumer will not just buy the car at the lowest price, but will look at other factors as well. The Hotelling model has been modified by many people, most notably d’Aspremont, Gabszewicz and Thisse (1979). Phlips and Thisse (1982) give an introduction on spatial models; Tirole (1988), Shy (1995) and Anderson, de Palma and Thisse (1992) discuss the basic model in a more formal manner.

Hotelling assumed there were two producers, who were located along a line (the length of which we shall normalise to 1), which could be seen as the Main Street of a town. Say that producer 1 is located at distance a from the far left of the line, and producer 2 is located at distance b from the far right of the line. Producer 1 and 2 set a price of p1

and p2, respectively. Consumers are uniformly distributed over the line, with unit density.

they get utility V from this. So, if a consumer located at x ∈ [0, 1] buys the good from producer 1, he gets a net utility of Ux = V − p1− |a − x|t.

In another interpretation this model becomes a model of product differentiation. Say that the line represents a certain characteristic of the good, for example the heat level of a curry (with the left representing a mild curry, and the right a very hot curry). Consumer locations can then be seen as the consumer’s ideal variety - so a consumer located at x = 0 will most prefer a mild curry. Transportation costs can then be seen as the disutility consumers derive from not being able to consume their most preferred variety. The analysis is the same for both interpretations.

Let us first consider the model in which producer 1 and 2 have fixed location a and 1 − b, and set prices p1 and p2, respectively. Consumers will buy from the producer that

will leave them with the highest net utility. A consumer is indifferent between the two producers if his location ˆx is such that

V − p1− |a − ˆx|t = V − p2− |(1 − b) − ˆx|t. (2.1)

This is the case for

ˆ

x = p1 − p2

2t +

1 − b + a

2 , (2.2)

where we have assumed without loss of generality that a ≤ ˆx ≤ (1 − b). The demand faced by producer 1 is now ˆ_{x, whereas the demand faced by producer 2 is 1 − ˆx. For this, we} have assumed that prices are such that the market is covered: all consumers will buy one unit of the good. The producers now maximise profits with respect to price. The optimal prices are

p1 = (3 − b + a)t

3 and p2 =

(3 + b − a)t

3 , (2.3)

and profits in equilibrium are

Π1 = (3 − b + a) 2_t

18 and Π2 =

(3 + b − a)2_t

18 . (2.4)

So, profits increase if transportation costs increase - this means that the more inaccessible are the stores, the higher are profits.

It can be shown that if both producers locate too close to each other, the equilibrium just presented does not hold. In this situation, producers will undercut each other’s prices, so that in this situation there is no equilibrium.4

Now let’s consider the case in which location choice is endogenous. Two opposing effects are important here. Firstly, each producer has an incentive to locate closer to his competitor, since

∂Π1

∂a > 0 and ∂Π2

∂b > 0. (2.5)

The reason for this is that, as a firm locates closer to his competitor, he gains more market share. This is called the demand effect. Secondly, if a producer moves closer to

4

his competitor this will intensify competition between them. This will lead the competitor to lower prices, which has a negative effect on the producer’s profits. This is called the strategic effect. The demand effect leads to less differentiation, whereas the strategic effect will lead to more differentiation.

In the case of linear transportation costs, the demand effect dominates the strategic effect, so producers want to locate as closely together as possible. This is called the principle of minimum differentiation. However, if they locate too close together there is no unique pure strategy price equilibrium. So, in the location and price game with linear transportation costs there is no equilibrium.

In order to fix this problem, d’Aspremont et al. (1979) consider a model with trans-portation costs that are quadratic in distance. So, a consumer located at x would now incur a transportation cost (a − x)2_{t if he bought the good from producer 1. It can be}

shown that the strategic effect dominates the demand effect in this case and producers will locate at the extremes of the line (see Tirole (1988) for details); so, in contrast to the minimum differentiation we get in the original Hotelling model, we now have maximum differentiation.

Lambertini (1994) considers a Hotelling duopoly with quadratic transportation costs in which firms are allowed to locate outside of the city. He finds that the demand effect and the strategic effect balance each other out outside the city boundaries, and both firms will locate symmetrically outside of the city. So, the result of d’Aspremont et al. (1979) that both firms locate on the edges is in fact a corner solution, caused by the restriction that firms have to locate inside the city boundaries.

The result of maximum differentiation depends heavily on the type of competition: if there is Cournot instead of Bertrand competition, then firms are found to agglomerate (Anderson and Neven 1991, Gupta, Pal and Sarkar 1997). This result seems more realistic, given the amount of agglomeration found in the real world (e.g. shopping malls, fashion labels that sell very similar items).

Another spatial model is the circular city model, introduced by Salop (1979). In this model the city is represented by a circle of unit circumference, with consumers distributed uniformly over it with unit density. A natural interpretation of this model is that of a 24 hour clock, on which firms can schedule their services (e.g. plane departure times). The model is often used to determine the amount of entry there will be in a market. The standard model involves identical firms who have a constant marginal cost of production, and a fixed cost. They are assumed to be spread out evenly over the circle, so that each firm faces the same market situation. Firms compete only with their direct neighbours. As in the standard Hotelling model, consumers incur a linear transportation cost when they buy the good. The number of firms in equilibrium is determined by the fixed cost.

2.2

Supply and demand uncertainty

about some aspects of the economy, for example about the magnitude of demand (in the case of uncertainty for producers), or about product characteristics or the magnitude of supply (in the case of uncertainty for consumers). We consider a model with consumer side uncertainty about the location of supply. We will now first discuss some models that deal with supply uncertainty, and then some interesting models that deal with demand uncertainty.

Turnovsky (1971) discusses the situation in which prices are fixed, but the quantity produced is random. Consumers can put in an order for a good with the producer, who will then produce this good. Though the producer will try his best to produce the exact amount ordered, the exact amount produced by the producer is random. This may lead to either a surplus or a shortage of the product. Consumers can choose whether or not to consume any surpluses. This will depend on their budget constraints. It is shown that as the uncertainty about the actual amount produced increases, planned demand will also increase. Also, demand for goods of which supply is certain is independent of the demand for goods of which supply is uncertain.

Another interpretation of this model is offered by Martin (1981). He considers the following situation: a firm orders inputs from an input supplier. There may be stochastic shocks determining the actual delivery of the inputs; when these shocks occur they will always be negative, so that there is never excess input supply. An example is the demand for labour by a firm: a firm hires a certain number of workers for its factories, but on any given day some of these workers may not show up due to illness or other reasons. As in Turnovsky (1971) Martin finds that demand for uncertain inputs is larger than it would have been if these inputs had been certain. So, for example, if workers would never get ill and always come to work, the firm would hire less workers than it would when employees did get ill. However, in contrast to Turnovsky (1971), Martin finds that the demand by the firm for inputs that are not stochastic is influenced by the level of uncertainty about the supply of other inputs. The extent of this influence depends on the production function. So, for example, if there is a chance that some workers may not show up for work, a firm may decide to buy more machines in order to substitute for this. Obviously whether the firm is able to do this depends on the degree of substitutability between capital and labour. Yet another version of this model is presented by Turnbull (1986). In his model input supply is dichotomous: either the entire order is delivered, or nothing is delivered. In contrast to the previous two studies, demand for the stochastic input is not necessarily increased. Also, in contrast with Martin demand for non-stochastic inputs is decreased in response to the uncertainty in one of the inputs.

demand area. This is in accordance with the standard practice, in which a firm must locate inside the city boundaries, but clearly it may not always be realistic. Consider for instance the situation of a monopolist located at θ − ε, where ε is some small positive number. This monopolist is likely to get at least some demand from consumers located close to the boundary.

Casado-Izaga (2000) considers a Hotelling duopoly with demand uncertainty in which producers are allowed to locate outside the city boundaries. Other than this, the setting is the same as in Harter. Casado-Izaga finds that demand uncertainty leads to more product differentiation (so firms locate further apart) both in the case of simultaneous entry and with sequential entry. Product differentiation in this case is greater than in Harter because firms can locate outside the city boundary and still receive some demand. In Harter’s (1996) model firms have an incentive to locate closer to the centre in order to ensure being within the city boundaries, and to receive positive demand. Also, welfare is lower under uncertainty than it would be under perfect information. This is because firms are excessively differentiated, which increases transportation costs for consumers.

A generalisation of Casado-Izaga (2000) is offered in Meagher and Zauner (2004). They consider a Hotelling duopoly in which consumers are uniformly distributed over the interval [M −12, M +

1

2], where M has a density function f (M ) with support [M , M ]. The authors

consider two different cases: first, the case in which prices are set after the demand loca-tion uncertainty is resolved (the ex post case), and second, the case where prices are set before the uncertainty is resolved (the ex ante case). Locations are always chosen before the uncertainty is resolved. It is especially interesting to look at the effect of increasing uncertainty, as measured by the variance of M , on the level of differentiation and on prices. The authors find that more uncertainty leads to more differentiation, both in the ex ante and the ex post case. Expected prices and profits also increase with the level of uncertainty.

2.3

Transparency

Market transparency reflects the degree to which consumers and producers are informed about important market characteristics. For producers this can involve information about competitors’ cost structure, prices and consumer tastes for example. For consumers market transparency can relate to things like information about prices, product characteristics and the level of service by producers. The main focus in the literature is price transparency, though transparency with respect to product characteristics is also discussed.

sales; say that these sales are announced in the newspaper or on TV, and that informed consumers are the ones who have seen these ads, and uninformed consumers have not.

A producer can follow two different strategies: he can aim to set the lowest price, and thus attract all the informed consumers and a share of the uninformed consumers; or, he can set a higher price and sell only to uninformed consumers. This way he will sell less units of the good, but he will make a higher profit per unit. Clearly, both strategies have their merits. Varian considers only symmetric equilibria in which all producers use the same pricing strategy. He finds that producers will use a mixed strategy in equilibrium, where they are most likely to set ‘extreme’ prices. So, a producer will typically set either a high price or a low price: intermediate prices occur less often. This is in accordance with real practice, in which producers normally set their regular price, and give large discounts in some periods.

Polo (1991) provides a transparency model in a spatial setting. There are two firms who are symmetrically located at fixed points along a line, and who compete in prices. A fraction ϕ of consumers is informed, whereas a fraction 1 − ϕ is uninformed. Whether a consumer is informed or uninformed is independent of his position on the line. There are two types of uninformed consumers: in addition to being uninformed about price, consumers can also be uninformed about product characteristics. However, these cases are assumed to be mutually exclusive: so, if consumers are uninformed about price, they are informed about characteristics, and vice versa. Even though uninformed consumers do not know the exact value of the unknown variable, they are assumed to know the market distribution of this variable.

Conclusions in this model depend on the type of information uninformed consumers have. If consumers are uninformed about prices, then the equilibrium price will be higher than under full information. This is due to the fact that uninformed consumers react only to the average price in the market - so if consumers are uninformed, price elasticity drops, which allows prices to rise. If, on the other hand, consumers are uninformed about product characteristics, then the equilibrium price falls below the full information level. This can be explained because if consumers are only informed about prices, they will buy the good with the lowest price. This intensifies price competition and forces producers to lower their prices.

An extension to this model is offered by Schultz (2004), who considers a Hotelling price setting duopoly in which producer locations are endogenous. To avoid corner solutions, producers are allowed to locate outside the market area. Again, a fraction ϕ of consumers is informed, the rest is not. In contrast to Polo’s (1991) model uninformed consumers are uninformed about both prices and product characteristics. Uninformed consumers are assumed to have rational expectations about prices and product characteristics. The equilibrium is assumed to be symmetrical, so that each producer gets half of the uninformed consumers.

causes prices to drop. An increase in transparency is beneficial to consumers, because they benefit from lower prices, and from being able to choose a better product (because the probability that a consumer buys a product that is not his favourite decreases with the level of transparency).

Boone and Potters (2002) consider a model where firms selling differentiated products compete in quantities. Consumers have a taste for variety: however, only a fraction of consumers is informed about all varieties and prices. The rest of the consumers is informed about the variety and price of only one seller. It is found that, as in Polo (1991) in the case where consumers are uninformed about varieties, prices may increase as transparency increases. This is because, as transparency increases, more consumers become aware of all the varieties. Since consumers have a taste for variety, this leads to an increase in demand, which allows prices to rise. This demand effect may offset the competition effect that causes prices to fall as transparency increases.

Our model will be closest to the one by Polo (1991), because we will assume consumers are informed about prices, but not about product locations. However, in the case of uncertainty the interpretation of the spatial model becomes important. Polo uses the interpretation of product differentiation, whereas we choose the literal interpretation of physical location of the good. In our model, if a consumer visits the wrong shop he will not be able to buy the good at all; in Polo’s (1991) he will be able to buy the good if he visits the ‘wrong’ shop, although he will receive a lower surplus. This obviously has consequences for the analysis.

2.4

Search models

Search models are similar to transparency models, in the sense that there is some market variable (usually price) that (some) consumers are uninformed about. However, instead of just visiting one producer and deciding whether or not to buy the product based on the one sample, consumers can visit multiple stores, at a positive search cost.

Burdett and Judd (1983) consider a model with a large number of identical firms. Consumers are assumed to know the distribution of prices, but are not informed about which firm sets which price. They may visit firms to get price quotations, at a positive search cost c. The first quotation is free. Consumers will buy at most one unit of the good from the firm who gave the lowest price quotation, as long as this price does not exceed the consumer’s reservation price for the good. Consumers search in a non-sequential manner: this means that they decide how many prices to sample before they start searching. It is found that an equilibrium in which all firms charge the monopoly price exists, as well as an equilibrium in which prices are dispersed, if search costs are not too high.

This model is modified somewhat in Stahl (1989). There are two main differences between the model by Stahl and the one by Burdett and Judd: firstly, Stahl lets consumers search sequentially instead of non-sequentially. This means that consumers decide, after visiting a firm, whether they want to sample another firm or not. So, if a consumer finds a satisfactory price at the first store he will stop the search immediately, whereas he might have sampled another firm in a non-sequential search setting. Secondly, Stahl assumes that a proportion ϕ of consumers can search at zero cost. This might be explained because some consumers enjoy the act of shopping, and hence do not get as much disutility from searching as other consumers. These consumers are called ‘shoppers’ by Stahl, and they are comparable to the informed consumers in the transparency models. Consumers who are not shoppers face a strictly positive search cost, but they do receive the first price quotation for free. To keep our terminology consistent we will call shoppers informed consumers and non-shoppers uninformed consumers, even though this may not be correct, strictly speaking (after all, non-shoppers can become informed by searching). Uninformed consumers will keep searching until they find a price that is no higher than their reservation price. In equilibrium it is optimal for producers never to set a price higher to this reservation price, so uninformed consumers will only search once.

As in Varian’s (1980) transparency model there are two strategies that firms may follow here: firstly, they may aim to be the lowest pricing firm, and thus attract all of the informed consumers, and some of the uninformed consumers. Secondly, they can decide not to be the lowest pricing firm, and aim only for the uninformed consumers. Which strategy is optimal depends on the proportion of informed consumers, and the number of firms. If ϕ = 1 then the model is equivalent to the Bertrand (1883) model, and prices are equal to marginal cost. If, however, ϕ = 0, then the Diamond (1971) result obtains: prices are set to the monopoly level.

increases, the price converges to the monopoly level. This can be explained by the fact that, as the number of firms increases, the probability of being the lowest pricing firm decreases. This makes it unattractive to lower price to attract the informed consumers, so producers will aim only at the uninformed consumers and set a higher price.

This model is generalised in Stahl (1996) to allow search costs to be heterogeneous. The main difference between Stahl (1996) and Stahl (1989) is that the monopoly price is never an equilibrium if there is a contingent of informed consumers, or if the distribution of search costs also includes zero search costs. So, in this case the monopoly price can only be an equilibrium when all consumers face strictly positive search costs. This contrasts with Stahl (1989), where the monopoly price could be an equilibrium even in the presence of informed consumers, if the number of firms was large enough.

Another generalisation of Stahl (1989) is offered by Janssen, Moraga-Gonz´alez and Wildenbeest (2004), who assume that the first price quotation is also costly (recall that in the previous models the first quotation was free). There are now two types of equilibria: a full participation equilibrium, in which all consumers search, and a partial participation equilibrium in which some of the uninformed consumers decide not to search because they expect prices to be too high. It is found that expected price increases as search costs decrease. This can be explained by the fact that lower search costs leads to more partici-pation. Because uninformed consumers will only search once, firms have monopoly power over uninformed consumers, allowing them to raise prices as more uninformed consumers enter the market. Another result is that an increase in the proportion of informed con-sumers does not affect the expected price. Also, in contrast to the result in Stahl (1989), the equilibrium price does not converge to the monopoly price if the proportion of informed consumers decreases to zero. This can be explained by the fact that, as prices rise, search becomes less attractive to uninformed consumers. So, if the price becomes too high, some consumers will not enter the market. This keeps prices from becoming too high. Using the same reasoning prices will not converge to the monopoly level if the number of firms becomes arbitrarily large.

of firms goes to infinity is equivalent to pricing in a duopoly. This is because it becomes increasingly unlikely for each firm to set the lowest price, so each firm will aim at the uninformed consumers. Each uninformed consumer will know at most two prices, so that each firm competes over each uninformed consumer with at most one other firm.

So far we have only discussed search models with homogeneous goods, but there is also a whole strand of search models which deals with differentiated products, often in a spatial setting. The first of these models is Wolinsky (1984), who considers a Salop (1979) model in which consumers are uninformed about which brands are on offer. There are many firms, who each sell a unique product at a fixed price p. Consumers’ willingness to pay depends on how far a product is removed from their ideal brand. In equilibrium, consumers will search until they find a brand that is located at an arc-distance from their most preferred brand of at most R. Wolinsky considers a free-entry equilibrium, and finds that there is excessive entry compared to what would be socially optimal. This may be explained by the fact that the optimal number of brands is bounded due to the limited information that consumers have. If there are more brands than this optimal number, consumers will not be able to benefit fully from this, because they may not be able to find the brand closest to them; this constitutes a social waste. However, as search costs decrease, the optimal number of firms increases, and the problem of excessive entry lessens.

Stahl (1982) considers a model with differentiated products, in which firms can lower search costs for consumers by locating closer together. Producers are assumed to compete in quantities in this model. Consumers are assumed to know prices, and the locations of market places (where a market place is the location of a firm, or of a cluster of firms), but not the composition of these market places. Also, consumers are assumed to know which varieties of the good are sold. Each consumer can visit at most one market place, and will choose that market place which gives the highest expected utility. It is found that there will be excessive concentration in this case, with all firms locating together with at least one other firm. Also, if consumers are choosy enough to only want one product, then all firms locating together will be the unique equilibrium. This is because in this case firms are not actually each other’s competitors, so they can locate closely together in order to attract more consumers.

Another model in which producers may choose to locate together is presented by Fischer and Harrington (1996). They consider a model in which firms can choose to agglomerate in a shopping mall, or locate separately in a periphery. They find that the degree of agglomer-ation increases with the level of product heterogeneity. If products are very heterogeneous, then competition between firms is less fierce. This makes locating together more attractive, because more consumers will come to an agglomeration if there are more firms in it. If an agglomeration receives more consumers, then each firm in it is likely to receive more consumers as well, which has a positive effect on profits.

what price, and search for the best price/quality combination. There are three possible equilibria: in the first, all firms sell the complex good, but some charge a high price, and others charge a low price to attract ordinary consumers. In the second, some firms sell the basic good, some the complex good, but all set the same low price. In the final equilibrium some firms sell the basic good at a low price, whereas others sell the complex good at a high price. So, in this model we find that the same quality good may be sold at different prices by different firms, and different qualities may be sold at the same price.

3

A model of supply location uncertainty

Now consider a Hotelling model with two firms. Each firm is located at either extreme of the line [0, 1]. Let’s assume that both firms are competitors in a certain product field (such as clothing), but that one of the firms is the single seller of a particular good (e.g. a certain type of pants); we will call this the specialty good. So, for this good that seller is the monopolist. Let’s assume that the firm located at x = 1 is the one selling the specialty good. From now on we will only focus on the market for the specialty good, and not look at interaction with goods that both producers may have in common. We do this to keep the model simple, and because it is likely there would not be a lot of interaction if the other goods sold by both producers are poor substitutes for the specialty good.

As in the standard model, consumers are uniformly distributed over the line, with unit density. We assume that all consumers have the same valuation V > 0 for the specialty good, and will buy at most one unit of this good. Consumers incur a linear transportation cost t per distance travelled when they visit the shops. This transportation cost can be seen as a return ticket: a consumer located at x wishing to visit the specialty good producer will incur a total transportation cost of t(1 − x), which will get him from his house to the shop, and back again. In principle if there were any other shops along the way, he could visit these without any extra cost. We assume that a fraction ϕ of consumers is informed about which shop sells the specialty good; the rest of the consumers is uninformed. Informed consumers will behave in the normal utility optimising fashion; uninformed consumers search in a non-sequential manner to find the good. These consumers face search costs in the form of added transportation costs if they do not find the good in the first shop they visit and decide to look further. Because of this, transportation costs may be seen as a search cost. Search costs in turn are an indirect measure of market transparency (where the proportion of informed consumers would be a direct measure of market transparency). We assume that consumers are informed about prices, even if they are not informed about the exact location of the good. This seems like a reasonable assumption, because consumers are fully informed about everything except the good’s location; since they are also perfectly rational it stands to reason that they are able to derive the price of the good for themselves.5 _{This differs from other search models where consumers do not know about}

prices, because in this case we have a monopolist instead of oligopolists. In an oligopoly situation consumers are often assumed to know the distribution of prices, which obviously becomes trivial in case of a monopolist. Furthermore, the focus in this paper is the effect

5

of search for a particular good, not the effect of search for the lowest price.

We will next solve the optimisation problem for consumers and derive the demand function. Then we will solve the producer problem and arrive at the market equilibrium. Finally we will look at the effects of changing the level of market transparency on consumer surplus, profits and prices. All proofs are in the appendix. Note that henceforth we will call the specialty good ‘the good’, and its producer ‘the producer’.

3.1

The consumer problem

Since the prior probability that the good is sold in any one shop is 1

2, we assume that

uninformed consumers will visit the shop closest to them first, because this yields the highest expected utility. There are now three types of consumers, each with distinct consumption patterns:

a) uninformed consumers with location x < 1/2 b) uninformed consumers with location x ≥ 1/2 c) informed consumers.

We will now look at each type of consumer in turn, discussing how they make their con-sumption decision. We will only outline this, details are in the appendix.

Case a)

There are three possibilities for these consumers: • No search: EUa

0 = 0

• Search once: EUa1 = 12(V − p) − tx

• Search twice: EUa

2 = V − p − 12t(1 + x),

where EUj_i is the expected utility of a consumer of type j who searches i times, and p is the price charged by the producer. Consumers of this type will go to the store located at x = 0 first, because this is the store closest to them. However, since the good is located at x = 1 they have to visit the second store as well in order to buy the good; so a condition for these consumers to buy the good is that they search twice. They will do this only if the expected utility of searching twice in non-negative (EUa_{2 ≥ 0) and the expected utility of} searching twice is at least as high as the expected utility of searching once (EUa

2 ≥ EUa1).

the good at the first shop, he will not be tempted to try the second because this will only decrease his utility. If he does find the good at the first shop he gets a positive pay-off (since V − p − tx ≥ 0 if 12(V − p) − tx ≥ 0). A similar argument holds for a consumer who

decides to search twice. So, our assumption that consumers search non-sequentially makes no difference: consumer behaviour is the same under sequential and non-sequential search.

-x p−V t + 1 2(V −p) t − 1 p−V t + 1 2(V −p) t − 1 x EUa 1 > EUa2 EUa 1 > EUa2 EUa1 > EUa2 EUa 1 < EUa2 EUa1 < EUa2 EUa 1 < EUa2 EUa 2 > 0 EUa2 > 0 EUa 2 > 0 EUa2 < 0 EUa2 < 0 EUa 2 < 0 p ≤ V −2 3t p > V −23t | {z } demand

Figure 3.1: Overview of expected utility in case a)

Figure 3.1 gives an overview of the values of expected utility for different locations (values of x). In those locations for which EUa

2 ≥ EUa1 and EUa2 ≥ 0 all consumers will buy

the good. In other locations demand will be zero. So, demand for the good by consumers of type a) will be equal to the distance over which EUa2 ≥ EUa1 and EUa2 ≥ 0. We can

derive from the figure that demand is zero for p > V − 2

3t. For lower values of p only

consumers located ‘in the middle’ will buy the good: those to the left will not find the second search profitable, and those to the right decide not to search at all because the expected transportation cost is too high. As p decreases the area for which consumers will buy the good increases, until all consumers buy the good.

Case b)

• No search: EUb0 = 0

• Search once: EUb1 = 12(V − p) − t(1 − x)

• Search twice: EUb

2 = V − p − t(1 − 12x).

Since consumers of this type are located closest to the shop at x = 1, this is the shop they will visit first if they decide to search. Since they will find the good in this shop, and buy it, searching at least once is enough for consumers of this type to buy the good. So the condition for consumption is that EUb_{1 ≥ 0 or EU}b_{2 ≥ 0. Note that if this condition} holds and a consumer decides to search, he will also want to buy the good once he finds it because then V − p − t(1 − x) ≥ 0. -x p−V 2t + 1 2(p−V ) t + 2 EUb₁ > 0 EUb₁ > 0 x EUb₁ > 0 EUb₁ < 0 EUb₁ < 0 EUb₁ < 0 EUb₂ < 0 EUb₂ < 0 EUb₂ < 0 EUb₂ > 0 EUb₂ > 0 EUb₂ > 0 p ≤ V −2 3t p > V −2 3t p−V 2t + 1 2(p−V ) t + 2 | {z } | {z } demand demand

Figure 3.2: Overview of expected utility in case b)

Figure 3.2 gives an overview of the signs of EUb₁ and EUb₂ for various locations of consumers. Consumers located in those areas where EUb

1 or EUb2 is non-negative will buy

Case c)

Recall that consumers of this type are informed about the location of the good. They will buy the good only if that gives them a non-negative utility, so if V − p − t(1 − x) ≥ 0. Demand

Some straightforward but tedious calculations (which are in the appendix) yield the fol-lowing total demand function:

Lemma 1 D(p) =              0 _{if p ≥ V} (1 + ϕ) V −p_{2t
if V −} 2_{3t ≤ p ≤ V} ϕ V −p<sub>t
+ (1 − ϕ)</sub>5(V −p)_{t − 3} _{if V −} 3 4t ≤ p ≤ V − 2 3t V −p t if V − t ≤ p ≤ V − 3 4t 1 _{if p ≤ V − t} (3.1)

We define the following price intervals: I0 = [V, ∞), I1 = [V −2₃t, V ], I2 = [V −3₄t, V − 2

3t], I3 = [V − t, V − 3

4t], and I4 = [0, V − t]. Note that these intervals overlap slightly.

However, since demand is continuous this imposes no problems here, or when profits are maximised, as we will see later. We can see that for p ∈ I0 demand will be zero, because

in that case the price of the good is higher than the valuation of the good. When the producer sets a price p ∈ I1 he will not attract any consumers of type a), and some of

type b) and c). In other words, he will only be attracting informed consumers, and some of the uninformed consumers located closest to him. If the producer charges p ∈ I2 he will

attract some consumers of all types. A price p ∈ I3 will attract all consumers of type b)

and some consumers of the other types. Note that with this price demand is exactly the same as it would be under perfect information. This does not mean however that there is no social waste in this situation: after all, consumers of type a) will still be spending too much on search costs, because they still visit the shop located at x = 0 first. When p ∈ I4

the market is covered and all consumers buy the good. Figure 3.3 shows the demand function for around ϕ = 1

2. Demand is continuous, but

the slope of the demand function is not the same everywhere (although can be verified that this is the case for ϕ = 1). The slope of the different parts of the demand curve is dependent on the group of consumers who buy the good in those parts. In the part of the demand curve belonging to I1 the slope is steep, because only uninformed consumers

close to x = 1 and informed consumers buy the good. The uninformed consumers decide to buy the good based on EUb

1, which increases only slowly in p. The demand curve for

I2 is not very steep. This can easily be explained by looking at Figure 3.1, and observing

that demand by consumers of type a) grows on two sides in this interval if p decreases. So, demand is highly reactive to changes in price here. The demand curve at I3 is a little

-6 D(p) p V V − 3 4t V − 2 3t V − t 1 3(1 + ϕ) 3 4 1

Figure 3.3: Plot of the demand function

3.2

The producer problem

Now that we have the demand function we can look at the producer problem. The producer will aim to maximize his profits:

max

p Π = D(p)(p − c), (3.2)

where c is the constant marginal cost of production. We assume that, as a first step, the producer decides how much of the market he wants to capture (and thus which price range he wants to be in), and maximises his profits according to this choice. For example, if a producer decides to focus on informed consumers and uninformed consumers located closest to him (at x ≥ 12), he will have to set a price in I1. He will then maximise his

profits using the part of the demand function belonging to p ∈ I1. If the optimal price

using this demand function is outside I1 there is a corner solution: if the optimal price

bound; if the optimal price is lower than the lower bound of the interval, the price will be set equal to the lower bound.

Some straightforward calculations yield the following optimal prices and profits: Lemma 2 Let p ∈ I1: For

V < c : pa 1 = V, Πa1 = 0 (3.3a) c ≤ V ≤ c +4₃t : pb₁ = 1 2(V + c), Π b 1 = (1 + ϕ)(V − c) 2 8t (3.3b) V > c +4 3t : p c 1 = V − 2 3t, Π c 1 = 1 3(1 + ϕ)  V − 2₃t − c  . (3.3c) Let p ∈ I2: For V < 11 − 7ϕ 15 − 12ϕt : p a 2 = V − 2 3t, Π a 2 = 1 3(1 + ϕ)  V − 2 3t − c  (3.4a) 11 − 7ϕ 15 − 12ϕt ≤ V ≤ c + 9 − 6ϕ 10 − 8ϕt : p b 2 = 1 2(V + c) − (3 − 3ϕ)t (10 − 8ϕ), Πb₂ = 1 t(5 − 4ϕ)  1 2(V − c) − (3 − 3ϕ)t 10 − 8ϕ 2 (3.4b) V > c + 9 − 6ϕ 10 − 8ϕt : p c 2 = V − 3 4t, Π c 2 = 3 4  V − 3₄t − c  . (3.4c) Let p ∈ I3: For V < c + 3 2t : p a 3 = V − 3 4t, Π a 3 = 3 4  V − 3₄t − c  (3.5a) c + 3 2t ≤ V ≤ c + 2t : p b 3 = 1 2(V + c), Π b 2 = (V − c) 2 4t (3.5b) V > c + 2t : pc_{3 = V − t,} Πc_{3 = V − t − c.} (3.5c) Let p ∈ I4: p4 = V − t, Π4 = V − t − c. (3.6) Note that Πc

1 = Πa2, Πc2 = Πa3 and Πc3 = Π4, and pc1 = pa2, pc2 = pa3 and pc3 = p4. So,

prices and profits are continuous over the various price intervals. This implies that our previous decision to make the intervals overlap slightly was innocuous.

In Lemma 2 the prices and profits with superscripts a and c are corner solutions, at the upper and lower bounds of p, respectively. If the producer decides to set the price equal to the upper bound of a price interval, he aims to just capture a certain group of consumers (if he set the price slightly higher he would lose some of those consumers). For example, if he sets price pa

3 he just captures all consumers located at x ≥ 12, and some of the other

For every configuration of parameter values, there are four potentially optimal prices and profits. For example, if V = c + 3t, then Πc

1, Πc2, Πc3 and Π4 are all possible. In order

to find the optimal solution, all profits that are feasible for given parameter values need to be compared. This tedious task is left for the appendix. Define the following intervals for V : I_{1 = [0, c) (3.7a)} I_{2 =}  c, c +2 3t + 2 3t r 1 + ϕ 10 − 8ϕ  (3.7b) I_{3 =}  c + 2 3t + 2 3t r 1 + ϕ 10 − 8ϕ, c + 9 − 6ϕ 10 − 8ϕt  (3.7c) I_{4 =}  c + 9 − 6ϕ 10 − 8ϕt, c + 3 2t  (3.7d) I_{5 =}  c + 3 2t, c + 2t  (3.7e) I_{6 = [c + 2t, ∞). (3.7f)}

Proposition 1 The optimal prices and profits are given by: if V ∈ I1 : p = pa1, Π = Π a 1 (3.8a) if V ∈ I2 : p = pb1, Π = Πb1 (3.8b) if V ∈ I3 : p = pb2, Π = Πb2 (3.8c) if V ∈ I4 : p = pa3, Π = Πa3 (3.8d) if V ∈ I5 : p = pb3, Π = Π b 3 (3.8e) if V ∈ I6 : p = p4, Π = Π4 (3.8f)

From Proposition 1 we can see that as the valuation of the good increases, the producer will aim at increasingly larger portions of the market. This can be explained because, as consumer valuation of the good increases, consumers will be more willing to travel further to find the good. This means that, all else being equal, demand increases as the valuation increases. Only in region I4 does demand stay the same as the valuation increases: this

is a transition range in which the producer sets a price to just attract all of the customers of type b) (and some of the consumers from the other groups as well). Note that I4 is

empty when ϕ = 1.

As can be seen in Figure 3.4 price generally increases with V , but at a lower rate. There is one discontinuity in price, at the transition between region I2 and I3. This also marks

the transition from setting a price in price aimed at consumers of type b) and c) (p ∈ I1)

to setting a price aimed at consumers of all three types (p ∈ I2). The fall in price can

be explained by a jump in the price elasticity: the transition from I1 to I2 brings about a

... .... .... ... ... .... .... .... .... .... .... ... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... .... .... .... .... .... .... .... .... ... ... .... .... .... .... .... .... .... .... .... .... .... .... .... . ... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... .... . V c + t p c + 3₄t c+1₃t+1₃t q 1+ϕ 10−8ϕ− 3−3ϕ 10−8ϕt c + 1₃t + 1₃tq_10−8ϕ1+ϕ c + _2(10−8ϕ)3 t | {z }| {z }| {z }| {z }| {z }| {z } I₁ I₂ I₃ I₄ I₅ I₆ c -6

Figure 3.4: Plot of optimal prices

attract more consumers, and make a higher profit. Note that this price jump disappears for ϕ = 1, so when all consumers are informed. In this case the demand function is also smooth, with no kinks, so there is no jump in price elasticity to induce the producer to lower his price.

3.3

Comparative statics and welfare

Consumer surplus

Consumer surplus is the sum of the net utility of all consumers. The net utility of a consumer located at x is

Ux = Ix(V − p) − transportation cost, (3.9)

where Ix = 1 if the consumer buys the good, and Ix = 0 if he does not buy the good. Note

that even if a consumer does not buy the good, any transportation cost incurred during the search for the good will be deducted from consumer surplus.

By deriving consumer surplus for each of the three types of consumers and adding up, we find the following result:

Lemma 3 Consumer surplus is given by

CS =                0 _{if p ≥ V} ϕ(V −p)_2t 2 _{+ (1 − ϕ)}(V −p)_4t 2 _{if V −} 2 3t ≤ p ≤ V ϕ(V −p)_2t 2 _{+ (1 − ϕ)}5(V −p)_2t 2 _{− 3(V − p) + t} _{if V −} 3 4t ≤ p ≤ V − 2 3t ϕ(V −p)_2t 2 _{+ (1 − ϕ)}(V −p)_2t 2 ₋ 1₈t _{if V − t ≤ p ≤ V −} 3₄t ϕ V − p −12t
+ (1 − ϕ) V − p −58t
if p ≤ V − t. (3.10) Varying ϕ

Now we will look at the effect that varying ϕ has on consumer surplus, profits and prices. In order to derive this effect, we need the partial derivatives of these quantities with respect to ϕ.

Starting with consumer surplus, it is straightforward to see the derivative with respect to ϕ is positive for p ∈ I2 and p ∈ {I4, I5}. For p ∈ I3 we have the following:

∂CS ∂ϕ = (V − p)2 2t −  5(V − p)2 2t − 3(V − p) + t  = 0 ⇐⇒ (3.11) (V − p) = 3₄t ± 1₄t. (3.12)

Because these zeros are outside the domain of p, and because ∂CS

∂ϕ is concave in (V − p) we

always have

∂CS

∂ϕ > 0. (3.13)

more consumers are informed, demand increases, which also has a positive effect on con-sumer surplus. It can also be shown that concon-sumer surplus of informed concon-sumers is higher than that of uninformed consumers.

We will now look at the effect of increasing transparency on profits. There are only two parts of the profit function that depend on ϕ:

Πb1 = (1 + ϕ)(V − c) 2 8t if V ∈ I2 (3.14) Πb₂ = 5 − 4ϕ t  1 2(V − c) − 3 − 3ϕ 10 − 8ϕt 2 if V ∈ I3. (3.15)

It is easy to see that Πb

1 is positively related to ϕ. The case of Πb2 is a bit more complicated:

∂Πb 2 ∂ϕ = − (V − c)2 t + 3 2(V − c) − 18(ϕ − 1)(2ϕ − 3)t (10 − 8ϕ)2 (3.16) = 0 ⇐⇒ (3.17) (V − c) = 3₄t _{1 ±} s 1 −8(ϕ − 1)(2ϕ − 3) (5 − 4ϕ)2 ! . (3.18)

These zeros can be shown to be on either side of I3, and since ∂Π

b 2

∂ϕ is strictly concave in

(V − c) we now have that

∂Πb 2

∂ϕ > 0, (3.19)

so that an increase in the proportion of informed consumers always has a non-negative effect on profits.

Looking at optimal prices, we find there is only one that depends on ϕ, namely pb 2 = 1

2(V − c) −

(3−3ϕ)t

10−8ϕ . It can be shown that this price monotonically increases to 1

2(V − c).

So, an increase in transparency has a non-negative effect on prices.

Concluding, we can say that despite the minor positive effect of increased transparency on prices, an increase in transparency is unambiguously good for total welfare. Both consumer surplus and profits rise with the proportion of informed consumers. This can be explained by the fact that, as more consumers become informed about the location of the good, demand increases, which has a positive effect both on consumer surplus and on profits (since price is systematically above cost).

Varying t

for uninformed consumers, which leads to more search, which means that in the end more consumers will be informed about the location of demand.

We begin by looking at consumer surplus. For most prices it is easy to see that consumer surplus and transportation cost are negatively related. This is not so easily shown though for p ∈ I3. For this range of prices we have the following:

∂CS

∂t =

−(5 − 4ϕ)(V − p)2

2t2 + (1 − ϕ), (3.20)

which is increasing in p. At the upper bound of the interval, p = V − 23t, (where the

derivative reaches its maximum over the domain) this derivative is equal to −19 − 1

9ϕ < 0.

We can now conclude that consumer surplus and transportation cost are negatively related for all prices. So, consumers are better off if the transportation cost (and thus search cost) falls.

Looking at profits, it can be seen that profits and transportation costs are negatively related. This means that profits rise if the transportation cost falls. This makes sense because with a lower transportation cost more of the informed consumers will find it worthwhile to buy the good, and more of the uninformed consumers will search for the good.

It can also be seen that prices fall as transportation costs rise. This is because if transportation costs are high, the producer does not have a lot of leeway to set prices in. If the transportation cost then falls, he can raise his price (though not by the same amount) without losing any customers. We arrive at the following proposition:

Proposition 2 Increasing the level of transparency, be it through increasing the propor-tion of consumers that is informed about the locapropor-tion of the producer, or through reducing transportation costs, has a positive effect on consumer surplus, profits and prices.

4

Conclusion

In this paper we have analysed a model in which there is consumer side uncertainty about supply location. Consumers are uniformly distributed over a line, and all want to buy a good from a monopolist known to be located at one extreme of this line. Some of the consumers are informed about which extreme of the line the producer is located at; the rest of the consumers is uninformed about this. We analysed consumer search behaviour and price setting behaviour by the producer. We also looked at the welfare effects of variations in the proportion of informed consumers, and of variations in transportation costs.

We found that relatively less of the uninformed consumers will consume the good than informed consumers. This is because the expected utility of uninformed consumers is generally less than that of informed consumers. Also, uninformed consumers that are located far away from the producer have to travel much further than informed consumers. This makes it less attractive for them to buy the good.

The producer sets his price depending on which part of the market he wants to capture. This in turn depends on consumers’ valuation of the good. For example, if consumer valuation is low, then uninformed consumers located far away from the producer will only buy the good if the price is very low. Since this may not be profitable for the producer he may decide to only aim at informed consumers and uninformed consumers living close to him.

the possible negative effect on the environment if more roads are built in an effort to reduce traffic jams, resulting in more people using their car instead of some other environmentally friendly mode of transportation. This negative effect on the environment could outweigh the positive effect on welfare that the decrease in transportation costs has.

We have kept to a very simple model to keep the analysis tractable. However, it would be very interesting to extend this model beyond the monopoly situation to make it more realistic. After all, there are usually substitute goods that consumers can buy if they cannot find the good they want. This may be modelled in much the same way as in Besancenot and Vranceanu (2004), in which there is a basic good and a complex good. Sticking to our Hotelling model with a firm at each extreme, we could assume that both firms sell the basic good, whereas only one firm sells the complex good. There are then two types of consumers: ordinary consumers who are indifferent between the two goods (or who value the complex good only marginally over the basic good), and expert consumers who will only buy the complex good because it has some feature they need. An example may be the mobile phone market: some phones do not have a camera (the basic good), and some phones do (the complex good). Most people will not really need a camera on their phone and may be willing to pay only a small amount more to get one. However, there is also a group of consumers (mostly teenagers probably) who will not want a phone without a camera (because otherwise their friends will think they are losers). It would be interesting to look at pricing and consumption strategies in this case, if some of the consumers do not know where to get the complex good.

Another possible extension is to use more firms, who may or may not sell the good. So, for example, there may be 4 firms in total of which only 2 sell the good. It could be interesting to see what happens if the proportion of firms that sells the good increases. Also, a different spatial design may be interesting, by placing shops on a circle instead of on a line. This would probably make the search dynamics more interesting because it would allow consumers to walk around in a circle and not have to backtrack. Also, consumers would be able to visit half the stores ‘for free’ because once they have covered half of the circle, they can choose which way they want to walk back home (recall that transportation costs are for a ‘return ticket’). Perhaps introducing an explicit search cost could be interesting as well. This search cost would represent the time a consumer has to spend in the shop looking for the good if he is not sure whether the shop sells it, or it may be a cost reflecting any anxiety a consumer might feel during the search (for example if the good is very important).

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A

Appendix

A.1

Proof of Lemma 1

Case a)

The expected utility of searching twice is non-negative if V − p − 1

2t(1 + x) ≥ 0

x ≤ 2(V − p)_t − 1. (A.1)

The expected utility of searching twice is at least as high as the expected utility of searching once if

V − p −1₂t(1 + x) ≥ 1₂V − 1₂p − tx

x ≥ p − V

t + 1. (A.2)

Comparing (A.1) and (A.2) to each other and to the domain of this case gives the following: p − V t + 1 ≥ 2(V − p) t − 1 ⇐⇒ p − V + t ≥ 2V − 2p − t ⇐⇒ p ≥ V −2 3t. (A.3)

Comparing (A.1) to the lower bound of x gives p − V

t + 1 ≥ 0

p ≥ V − t, (A.4)

Comparison of (A.2) to the lower and upper bound of x gives, respectively, 2(V − p) t − 1 ≥ 0 p ≤ V −1 2t (A.6) and 2(V − p) t − 1 ≤ 1 2 p ≥ V −3 4t. (A.7)

With Figure 3.1 we can easily derive the demand function for consumers in category a). Note that always p ≤ V − 1

2t if p ≤ V − 2 3t. Da_{(p) =}          0 _{if p ≥ V −} 2₃t (1 − ϕ)3(V −p)t − 2  if V −34t ≤ p ≤ V − 2 3t (1 − ϕ) V −pt − 1 2
if V − t ≤ p ≤ V −34t (1 − ϕ)1 2 if p ≤ V − t. (A.8) Case b)

In this case a consumer will buy the good if EUb_{1 ≥ 0 or EU}b_{2 ≥ 0. This gives the following} conditions: EUb₁ = 1 2(V − p) − t(1 − x) ≥ 0 x ≥ p − V 2t + 1 (A.9) and EUb 2 = V − p − t(1 − 1 2x) ≥ 0 x ≥ 2(p − V ) t + 2. (A.10)