A model of sales with a large number of sellers

Mathematical Social Sciences 104 (2020) 68–70

Contents lists available at ScienceDirect

Mathematical Social Sciences journal homepage: www.elsevier.com/locate/mss

Short communication

A model of sales with a large number of sellers✩ Evangelos Rouskas Section Microeconomics, Department of Regional and Economic Development, Agricultural University of Athens, Greece

article

info

Article history: Received 24 October 2018 Received in revised form 15 November 2019 Accepted 7 February 2020 Available online 13 February 2020 Keywords: Sales Large number of sellers Search Consumer heterogeneity

a b s t r a c t In this paper, I propose a model that can support Varian’s (Varian, 1980) equilibrium search behavior with an arbitrarily large number of sellers, even when the first price observation is costly for the consumers with positive search costs and the search is endogenous. In my model, the consumers with zero search costs have the same low valuation for all sellers’ products, whereas the consumers with positive search costs (i) learn the price and their valuation for the product of each seller after engaging in costly search for the corresponding seller; and (ii) in the pre-search phase, anticipate that the valuation for the product of each seller is high with strictly positive and lower than unity exogenous probability and low with the remaining probability. Conditional on that some reasonable restrictions on the parameters are satisfied, although the number of sellers may grow arbitrarily large, the expected price is bounded above by the low valuation, and the consumers with positive search costs find it most profitable to search once. © 2020 Elsevier B.V. All rights reserved.

1. Introduction The equilibrium in Varian’s model of sales (Varian, 1980) is characterized by the search behavior whereby consumers with zero search costs observe all prices in the market, whereas consumers with positive search costs observe one price with probability one by visiting a seller at random. Janssen and MoragaGonzález (2004, Proposition 3(ii)) showed that this search behavior cannot be sustained in equilibrium with an arbitrarily large number of sellers if the first price observation is costly for the consumers with positive search costs, and the search is endogenous. The rationale for this nonexistence result is that the expected price is increasing monotonically with the number of sellers, and, consequently, even when the search cost is very low, there exists a critical number of sellers above which the expected price is prohibitively high for the consumers with positive search costs. In this paper, I present a version of Varian’s model with endogenous search that does not give rise to the aforementioned nonexistence result. My key assumption is that, in the pre-search phase, each consumer with positive search costs faces uncertainty with regard to the valuation for the product of each seller; the valuation is high with strictly positive and lower than unity exogenous probability and low with the remaining probability. Whereas the valuation of the consumers with zero search costs ✩ I thank the Editor Juan D. Moreno-Ternero, an Associate Editor, and a referee for their helpful attitude. E-mail address: [email protected]. https://doi.org/10.1016/j.mathsocsci.2020.02.002 0165-4896/© 2020 Elsevier B.V. All rights reserved.

is always low.1 In my version, for some parameters, the expected price does not exceed the low valuation, irrespective of the number of sellers. This can render searching one price optimal for the consumers with positive search costs, even though the number of sellers may grow arbitrarily large. For instance, consider a real-world online platform where consumers enter to buy an item. The consumers may differ in that some of them may rate highly, for example, either the quick delivery time or the very good packaging of the item. Those consumers with either time-delivery or packaging concerns will enjoy a high gross benefit if they buy the item from a seller that satisfies their requirements, and will receive a low gross benefit otherwise. To find out which seller will meet their preferences in the best way, the consumers can consult online reviews for each seller. Hann and Terwiesch (2003) showed that there exist frictional costs in online environments. This means that the process of searching online for the price of each seller and consulting online reviews for each seller may be costly. My theoretical setting analyzes such situations. Literature The present paper belongs to the literature on consumer search. The reader is referred to Baye et al. (2006) and Anderson and Renault (2018) for surveys of this literature. Ding and Zhang (2018) adopt a similar—at least in certain aspects— framework. These authors assume that some consumers know their valuation for the product of each seller, whereas some other 1 As higher income can be associated with higher opportunity cost of time and higher willingness to pay, here, the consumers with positive search costs can be viewed as consumers with higher income compared to the consumers with zero search costs.

E. Rouskas / Mathematical Social Sciences 104 (2020) 68–70

consumers have to engage in costly search to learn their valuation. Furthermore, with some probability, the valuation for the product of each seller is high, and with the remaining probability the valuation is low.2 The paper proceeds as follows. Section 2 presents the model, Section 3 provides the analysis, and Section 4 concludes. The proofs are presented in the Appendix. 2. The model In the static economy under examination, all agents are risk neutral and have a common outside payoff equal to zero. I consider a general supply structure with N ≥ 2 sellers. The sellers supply a single product and compete in prices. All sellers have the same constant marginal cost, which equals zero. There are various distinct variants of the single product, and each seller supplies only one variant. The consumers are nonatomic, with total mass equal to one, and have unit demand. There exist two types of consumers. A percentage λ ∈ (0, 1) of consumers observes the prices of all sellers without having to incur any search cost and enjoys the same gross benefit vL > 0 from all variants. The consumers with zero search costs purchase the product from the seller with the lowest price in the market if that price is weakly lower than vL . If two or more sellers provide the lowest price in the market, the consumers with zero search costs select each one of these sellers with the same probability. The remaining percentage of consumers, 1 − λ, has to incur a strictly positive search cost to learn simultaneously both the price and the gross benefit of a seller’s variant. The consumers with positive search costs engage in nonsequential search, i.e., they decide how many sellers they will search before learning the prices and the gross benefits, and they enjoy different gross benefit from different variants; some variants deliver a gross benefit equal to vL and other variants deliver a gross benefit equal to vH with vH > vL . Each consumer with positive search costs anticipates that each seller will supply a high-gross-benefit variant with probability x ∈ (0, 1) and a low-gross-benefit variant with probability 1 − x. I focus only on equilibria wherein the consumers with positive search costs search one seller with probability one and choose a particular seller with probability N1 . I also direct attention only to parameters satisfying Assumption 1: v

Assumption 1 (x < v L ). This assumption gives no incentives to H sellers to set prices strictly higher than vL . The reason is that a (1−λ)vL seller can secure profits when the maximum price ever N x(1−λ)vH charged equals vL and profits when the maximum price N ever charged exceeds vL . Of these two levels of profits, the former is strictly higher if Assumption 1 holds. The search cost for the first seller sampled is represented by the parameter c ∈ (0, x(vH − vL ) + vL ). Note that, for c > x(vH − vL ) + vL , even if all prices in the market are equal to the marginal cost, the payoff for the consumers with positive search costs from searching one seller with probability one is strictly lower than the outside payoff. To simplify the procedure of endogenizing the search behavior, I impose the restriction that the search cost for a seller beyond the first is strictly higher than vH . 2 The model presented here generates the prediction that an increase in the percentage of consumers with zero search costs leads to a decline in the prices—to the degree that the focus is on the oligopolistic supply structure. This holds true even though the present context encompasses product differentiation. Arguably, an increase in the percentage of consumers with zero search costs represents an increase in market transparency. In related work on Bertrand oligopolies with product differentiation, Cosandier et al. (2018) demonstrate that following an increase in market transparency one of the prices may increase.

69

The detailed timing of the game follows Janssen and MoragaGonzález (2004). Sellers announce prices and consumers choose their search intensity simultaneously. The solution concept is that of Nash equilibrium. Denote θ1 the probability that the representative consumer with positive search costs incurs the search cost once. Each seller optimally chooses a price conditional on the price distributions of the rival sellers and θ1 . Each consumer with positive search costs optimally chooses θ1 conditional on his beliefs about the distributions of prices. I focus only on symmetric equilibria. Let F (p) denote the degenerate or nondegenerate distribution of prices charged and π (p) denote the per seller profits attained. An equilibrium is represented by {F (p), θ1 } which satisfies (a) given θ1 , π = π ∀p in the support of F (p), where π is a constant, and π ≤ π ∀p outside the support of F (p); and (b) θ1 is optimal, given that the beliefs by the consumers with positive search costs about the distributions of prices turn out to be correct. 3. Analysis I start the analysis by highlighting, through Proposition 1, how the pricing decisions of the sellers are determined when N ≥ 2. Due to Assumption 1, the cumulative distribution function defined in Proposition 1 is the same that would pertain to a market where both consumers with zero search costs and consumers with positive search costs have valuation vL . Proposition 1 also shows how the search behavior is endogenized. In detail, for θ1 = 1 to be optimal, θ1 = 1 must deliver a payoff which is strictly higher than the outside payoff (see Inequality (1)).3 Proposition 1. There exists a unique symmetric endogenous search equilibrium in which θ1 = 1, and, in this equilibrium, all sellers set prices according to the atomless cumulative distribution function F (p) = 1 −

(

(1−λ)(vL −p) λNp

) N −1 1

(1−λ)v

with support [ 1+(N −1)L λ , vL ]. This

equilibrium exists conditional on that Assumption 1 and Inequality (1) hold x(vH − vL ) + vL − E(p) > c

(1)

where E(p) represents the expected price. In line with Janssen and Moraga-González (2004, Proposition 3(i)), I argue that the expected price that is derived from the cumulative distribution function defined in Proposition 1 is increasing monotonically in the number of sellers. As the number of sellers grows arbitrarily large, the difference vL − E(p) becomes negligible, and, in contrast to Janssen and MoragaGonzález (2004, Proposition 3(ii)), θ1 = 1 can be optimal.4 This is emphasized by Proposition 2. Proposition 2. Fix a constellation of parameters {vH , vL , x, λ, c } which satisfies Assumption 1. Whenever both under {vH , vL , x, λ, c , ˜ N } and under {vH , vL , x, λ, c , ˜ N + 1} Inequality (1) is also satisfied, 3 Suppose the consumers with positive search costs were allowed to search, for example, two sellers, and both sellers in the sample would provide a lowgross-benefit variant with probability (1 − x)2 and so on. Then, θ1 = 1 should also deliver a strictly higher payoff than is the payoff generated from searching two sellers. Some complexity arises in cases when a consumer with positive search costs who searches two sellers enjoys different gross benefit from the different sellers, which would occur with probability 2x(1 − x). In such cases, the consumer in question observes both sellers’ gross benefits and prices and purchases the product from the seller offering the higher net benefit, given the actual price realizations. To circumvent this complexity, one could restrict attention to equilibria which satisfy the condition vH − p > vL − p where p and p represent the upper and the lower bound, respectively, of the set from which the sellers randomly select prices. Then, a consumer who enjoys different gross benefit from two sellers selects the seller with the high gross benefit. 4 Note that a direct analysis of the perfectly competitive structure is not well-defined.

70

E. Rouskas / Mathematical Social Sciences 104 (2020) 68–70

then E(p; ˜ N + 1) > E(p; ˜ N). Moreover, whenever Assumption 1 and Inequalities (2)–(3) are satisfied simultaneously vL c < (2)

vH − vL

vH

c

x>

(3)

vH − vL

then the equilibrium described in Proposition 1 is valid even for a supply structure with an arbitrarily large number of sellers that generates an expected price arbitrarily close to vL .

In equilibrium, the expected price equals

I have shown that Varian’s (1980) assumed search behavior can be supported in a model with endogenous search even when the number of sellers grows arbitrarily large and the first price quote is costly for the consumers with positive search costs. The assumption that drives this result is that the consumers with positive search costs are uncertain about the valuation for the product of each seller in the pre-search phase. The valuation can be either high or low. In this environment, as the number of sellers grows arbitrarily large, it is possible that the expected price does not exceed the low valuation, and the consumers with positive search costs find it optimal to search exactly one seller.

Proof of Proposition 1. The upper bound of the support of the price distribution must be weakly lower than vH ; otherwise pricing at the upper bound yields zero profit due to zero demand. No seller has an incentive to set a price strictly higher than vL . Suppose to the contrary that the upper bound is strictly higher than vL . Then, pricing at the upper bound does not attract any consumers with zero search costs and yields a level of profit x(1−λ)vH . Instead, pricing at vL yields a level of weakly lower than N (1−λ)v (1−λ)v profit at least equal to N L . Assumption 1 means that N L > x(1−λ)vH . This constitutes a contradiction. Hence, each seller can N (1−λ)vL . The expected profit of a seller who secure profits equal to N charges p (≤ vL ) when the rival sellers set prices according to the cumulative distribution function F (p) and each consumer with positive search costs incurs the search cost once with probability one is:

with probability one is x(vH − vL ) + vL

+ p(1 − F (p))N −1 λ N This expected profit is analyzed as follows. The seller captures the consumers with positive search costs, i.e., a mass equal to 1 − λ, with probability N1 , even if p is the highest price in the market. The seller captures an additional mass equal to λ, i.e., all the consumers with zero search costs, if p is the lowest price in the market, which occurs with probability (1 − F (p))N −1 . In equilibrium, the sellers play no atoms (see Janssen and MoragaGonzález, 2004 Proof of Lemma 2), and the expected profits must (1−λ)vL be equal to , that is: N + p(1 − F (p))N −1 λ =

(1 − λ)vL

(1−λ)v

(A.1)

N

From Eq. (A.1) it is derived that F (p) = 1 −

(

(1−λ)(vL −p) λNp

) N −1 1

with support [ 1+(N −1)L λ , vL ], where the lower bound of the support p satisfies F (p) = 0. The support has no gaps (see Varian, 1980, Proof of Proposition 8). Next, I use the following change of variables:

(

1−

c. □

vL

1

∫

(

vL

(1 − λ)(vL − p)

λNp

) N −1 1

(˜ N +1)λ ˜ zN 1−λ

1+

0 1

∫

1

⎛

λ 1−λ

⎝(

1+

0

∫1

1 dz 0 1 + N λ z N −1 1−λ

−

− 1+

vL

1

z N −1 (˜ N − (˜ N + 1)z)

λ

⎝(

1+

˜ N ˜ N +1

∫

)(

1+

˜ Nλ ˜ z N −1 1−λ

∫ −vL

(˜ N +1)λ ˜ zN 1−λ

λ

1+

)(

1+

˜ Nλ ˜ z N −1 1−λ

vL

0

−vL

⎝(

(˜ N +1)λ ˜ zN 1−λ

)(

1+

˜ Nλ ˜ z N −1 1−λ

1

λ ˜ N + 1)z − ˜ N) z N −1 ((˜ 1−λ )( ) ⎠ dz ˜ (˜ N +1)λ ˜ N λ N −1 + 1−λ z N 1 + 1−λ z ˜

1+

≥

λ ˜ ⎟ z N −1 (˜ N − (˜ N + 1)z) 1−λ ⎟ )˜N ) ( )˜N −1 ) ⎠ dz ( ( ˜ ˜ ˜ (˜ N +1)λ Nλ N N 1 + 1−λ ˜ ˜ 1−λ N +1 N +1

⎞

1

⎜ ⎜( ⎝

1+

λ ˜ ⎟ z N −1 ((˜ N + 1)z − ˜ N) 1−λ ⎟ )˜N ) ( )˜N −1 ) ⎠ dz ( ( ˜ ˜ ˜ (˜ N +1)λ Nλ N N 1 + 1−λ ˜ ˜ 1−λ N +1 N +1

λvL

(

) ⎠ dz

⎞

⎜ ⎜( ⎝

˜ N ˜ N +1

⎞

˜

⎛ ∫

) ⎠ dz =

N − (˜ N + 1)z) z N −1 (˜

⎛ ˜ N ˜ N +1

∫

⎞

⎞

⎛

˜ N ˜ N +1

are positive and strictly

˜

1−λ

1

) ⎠ dz

˜ Nλ ˜ z N −1 1−λ

z N −1 (˜ N − (˜ N + 1)z)

⎛ ⎝(

0

⎞

˜

(˜ N +1)λ ˜ zN 1−λ

1−λ 0

vL

⎛

dz =

˜ Nλ ˜ z N −1 1−λ

(˜ N +1)λ ˜

∫

)

1

As both 1 + 1−λ z N and 1 + increasing in z, then

(1 − λ)p

z=

In-

equality (1) ensures that, for each consumer with positive search costs, incurring the search cost once with probability one yields a strictly positive payoff, i.e., a payoff which is strictly higher than the outside payoff. Observe that the payoff for each consumer with positive search costs from incurring ( the search cost once )

Appendix. Proofs

N

dz. 0 1+ N λ z N −1 1−λ

Proof of Proposition 2. It holds that E(p; ˜ N + 1) − E(p; ˜ N) =

4. Conclusion

(1 − λ)p

vL

∫1

(1 − λ) 1 +

˜ ˜ λ NN ˜ 1−λ (˜ N +1)N −1

=

1

∫

z N −1 (˜ N − (˜ N + 1)z)dz = 0. □ ˜

)2

0

References Anderson, Simon P., Renault, Régis, 2018. Firm pricing with consumer search. In: Corchón, Luis C., Marini, Marco A. (Eds.), Handbook of Game Theory and Industrial Organization, Volume II, Applications. Edward Elgar, Cheltenham, UK, pp. 177–224. Baye, Michael R., Morgan, John, Scholten, Patrick, 2006. Information, search, and price dispersion. In: Hendershott, Terrence (Ed.), Economics and Information Systems. Elsevier, Amsterdam, Netherlands, pp. 323–376. Cosandier, Charlene, Garcia, Filomena, Knauff, Malgorzata, 2018. Price competition with differentiated goods and incomplete product awareness. Econom. Theory 66 (3), 681–705. Ding, Yucheng, Zhang, Tianle, 2018. Price-directed consumer search. Int. J. Ind. Organ. 58, 106–135. Hann, Il-Horn, Terwiesch, Christian, 2003. Measuring the frictional costs of online transactions: The case of a name-your-own-price channel. Manage. Sci. 49 (11), 1563–1579. Janssen, Maarten C.W., Moraga-González, José Luis, 2004. Strategic pricing, consumer search and the number of firms. Rev. Econom. Stud. 71 (4), 1089–1118. Varian, Hal R., 1980. A model of sales. Amer. Econ. Rev. 70 (4), 651–659.