Equilibrium Price Dispersion with Search Regret Disutility

Abstract

I examine an extension of the Burdett and Judd ([1983]. Equilibrium price dispersion. Econometrica, 51[4], 955–970) model whereby the consumers with positive search costs experience search regret disutility. First, I focus on the non-sequential search equilibrium in which the said consumers randomize between searching for one price and searching for two prices. When the disutility is significant (a) the spectrum of parameters for which this dispersed price equilibrium can be sustained widens significantly compared to the setting with no disutility; (b) this dispersed price equilibrium is unique and stable in contrast to the multiplicity of dispersed price equilibria of this type which arise in the original model; and (c) in the stable dispersed price equilibrium of this type the consumers with positive search costs respond to the possibility of search regret disutility by increasing their equilibrium search intensity. Second, I concentrate on the noisy sequential search equilibrium in which the reservation price is endogenous. When the search cost takes relatively high values, then compared to the setting with no disutility (a) the set of parameters for which this dispersed price equilibrium is supported may become significantly smaller; (b) the reservation price decreases; and (c) the consumers with positive search costs choose the same search intensity.

JEL Classifications: D41, D83

Keywords

Full/partial search regret perfect competition disutility price dispersion

Introduction

Burdett and Judd (1983, hereinafter BJ83) in their seminal theoretical contribution presented a dispersed price equilibrium with non-sequential search. In this equilibrium, consumers randomize between searching for one price and searching for two prices and always purchase the product under examination with probability one after incurring the search costs. However, there is always positive probability that the price that will be paid will be slightly lower than the willingness to pay, and, consequently, only a part of the search-cost expenditures will be recouped indirectly through this positive surplus captured by consumption. In other words, BJ83 generate endogenously probabilistic search regret. It is possible that in hindsight the consumers find themselves in a position in which they would have been better off had they avoided entirely the procedure of search in the first place. ¹

In the original BJ83 formulation all consumers have positive search costs, whereas there exists a continuum of firms. Janssen and Moraga-González (2004) introduced consumers with zero search costs into BJ83 and examined a general supply structure. Even with consumer heterogeneity, though, the equilibrium wherein the consumers with positive search costs randomize between searching for one price and searching for two prices always generates endogenously probabilistic search regret.

BJ83 also presented a dispersed price equilibrium with noisy sequential search. In this equilibrium, consumers endogenize their reservation price, and by incurring the search cost once they learn one price with some probability, two prices with some probability, and so on. Contrary to the non-sequential search approach, the noisy sequential search approach does not always generate endogenously probabilistic search regret. On the one hand, when the search cost is relatively low, the endogenous reservation price is low as well, and there is no search regret. On the other hand, when the search cost is relatively high, the endogenous reservation price is high as well, and search regret occurs probabilistically. (The reader is referred to Appendix B for details on the noisy sequential search benchmark).

In this paper, I examine an extension of the BJ83 model whereby search regret causes a negative emotion to the consumers which is measured by a parameter as disutility. I analyse the impact of search regret disutility both on the BJ83 non-sequential search equilibrium and the BJ83 noisy sequential search equilibrium. In my model, I assume a perfectly competitive supply structure. Throughout this paper, consumers with positive search costs and consumers with zero search costs may coexist.

As far as the BJ83 non-sequential search equilibrium is concerned, I find the following results when the disutility is significant: (a) the spectrum of parameters for which this dispersed price equilibrium can be sustained widens significantly compared to the setting with no disutility (see Tables 1 and 2); (b) this dispersed price equilibrium is unique and stable in contrast to the multiplicity of dispersed price equilibria of this type which arise in the original BJ83 model (see Tables 1 and 2); and (c) in the stable dispersed price equilibrium of this type the consumers with positive search costs respond to the possibility of search regret disutility by increasing their equilibrium search intensity (see Tables 1 and 2). The latter result holds even when the disutility is not significant (see Tables 1 and 3) and means that both the profits and the social welfare decrease when the emotions of consumers are taken into account. Intuitively, in the stable non-sequential search equilibrium, the consumers with positive search costs attempt to decrease the probability of search regret as the search regret disutility increases. This is accomplished by an increase in the search intensity, which also increases the net pay-off of the consumers with positive search costs (see Table 4).

With regard to the BJ83 noisy sequential search equilibrium, it should be stressed that even in the presence of search regret disutility, probabilistic search regret does not occur for all values of the search cost. Therefore, in my treatment, I emphasize the high values of the search cost that lead to probabilistic search regret (see Assumption 1). By introducing search regret disutility to the BJ83 noisy sequential search environment (a) the set of parameters for which the noisy sequential dispersed price equilibrium is supported may become significantly smaller (see Tables 5 and 6); (b) the reservation price decreases (see Table 7); and (c) the consumers with positive search costs search once. As consumers with positive search costs use the endogenous reservation price rule, any firm that charges a price higher than the reservation price captures zero demand. In equilibrium, no firm will ever charge a price higher than the reservation price, similarly to the standard BJ83 noisy sequential search set-up. The reservation price decrease in response to the possibility of search regret disutility can be brought into parallelism with the increase in the search intensity in the non-sequential search environment. In the sequential search environment, the consumers with positive search costs again attempt to decrease the probability of search regret and bring about the desired result through the decrease in the reservation price. Importantly, any increase of the value of the search regret disutility parameter, either from a zero level to a positive level or from a nonzero level to a higher level, decreases the reservation price. This leads unambiguously to lower expected prices and lower probability of search regret; however, it is not clear whether the consumers with positive search costs are better off or worse off. Simulations suggest that the net pay-off of the consumers with positive search costs may either increase or decrease when the search regret disutility parameter increases (see Table 7). The analytical treatment of the BJ83 noisy sequential search model with search regret disutility presents technical difficulties. In the environment with search regret disutility, the noisy sequential search equilibrium is not supported for all values of the search cost for which it is supported in the environment without disutility (see Tables 5 and 6).

Literature

The present paper is related to the literatures on regret and on consumer search. An analytical treatment of search regret has not emerged hitherto in the interdisciplinary literature on regret.²

The notion of regret, broadly defined, has attracted great attention as it appears already in the epic work Odyssey attributed to Homer (Elster, 1984). Notably, the notion of regret has been connected to the notion of commitment. The decision-maker who anticipates that in the future he will engage in an action which will lead to regret had better commit himself not to be able to engage in this action (Schelling, 1984). Overall, there is significant research on regret that spans marketing, management and economics. For instance, see Filiz-Ozbay and Ozbay (2007), Syam, Krishnamurthy, and Hess (2008), Shih and Schau (2011), Nasiry and Popescu (2012), Jiang, Narasimhan, and Turut (2017), and the references therein.

Therefore, my work complements the existing literature on regret and can be best viewed as a follow-up to Reynolds, Folse, and Jones (2006). Reynolds et al. (2006) define search regret as the negative emotion experienced by a prospective consumer who is willing to purchase a certain product and after engaging in costly pre-purchase search for this product purchases nothing or a substitute. Although search regret was explicitly defined for the first time by Reynolds et al. (2006), as already mentioned, it was implicitly treated by BJ83. To make a distinction between Reynolds et al. (2006) and BJ83, I provide a generalization of the definition of search regret. To this end, I introduce new terminology: full search regret and partial search regret. Full search regret applies to all cases in which the prospective consumer either chooses the no-purchase option or purchases the product at a price which exactly equals the maximum willingness to pay after incurring pre-purchase search costs. In this type of search regret, the consumer regrets the total search costs incurred. Partial search regret applies to all cases in which the consumer purchases the product after incurring pre-purchase search costs, but the surplus generated by the consumption covers only a part of the total search costs. In this type of search regret, the consumer regrets only a part of the total search costs incurred. According to my terminology, Reynolds et al. (2006) deal with full search regret, whereas BJ83 generate endogenously probabilistic partial search regret.³

A duopolistic version of BJ83 can be modified to generate both probabilistic full search regret and probabilistic partial search regret. Such a modification would have the assumptions that there exists consumer heterogeneity in regard to search costs and non-sequential search is ordered. In more detail, when the consumers with positive search costs search once, a particular firm is selected. In the dispersed price equilibrium, this particular firm chooses the price that equals the maximum willingness to pay with positive probability. See Wilson (2010) for a sequential search model with ordered search.

The literature on consumer search is comprehensively surveyed by Baye, Morgan, and Scholten (2006) and Anderson and Renault (2018). In this literature, the possibility of search regret and the disutility it may cause have been overlooked, although search regret occurs with positive probability, for example, in all types of the non-sequential search equilibria studied by Janssen and Moraga-González (2004). Similarly, to one of my results, Galeotti (2010) finds a unique dispersed price equilibrium in a duopolistic non-sequential BJ83 framework with social networks.

This paper proceeds as follows. The next section presents the main assumptions of the model, the fourth and fifth sections provide the analysis and the sixth section concludes the paper. The proofs of the Propositions of the main body of the paper are presented in Appendix A. Appendix B presents a variant of the BJ83 noisy sequential search model with no search regret disutility.

The Model: Main Assumptions

Let N denote the number of firms. The industry is perfectly competitive, that is, $N \to \infty$ , and the setting is static. The firms supply a single product, engage in price competition, are risk neutral and have a common outside pay-off equal to zero. Each firm has the same constant marginal cost which is normalized to zero. Consumers are non-atomic and risk neutral with total mass equal to one. All consumers have unit demand; each consumer is willing to pay up to $v > 0$ for the product. A percentage $λ \in [0, 1)$ of the consumers has zero search costs, and the remaining percentage, $1 - λ$ , must incur a positive search cost $c \in (0, v)$ to learn a seller’s price. The no-search option delivers a pay-off equal to zero. Let $γ \geq 0$ represent the disutility that the consumers with positive search costs experience due to search regret. The parameter γ is equivalent to a lump sum expenditure.

Non-sequential Search

In the non-sequential search approach, each consumer with positive search costs must decide how many prices will be observed before observing any of them. The analysis is focused on the BJ83 dispersed price equilibrium whereby the consumers randomize between searching for one price and searching for two prices.⁴

Janssen and Moraga-González (2004) have shown that in the BJ83 non-sequential search equilibrium with consumer heterogeneity, the cumulative distribution function according to which the firms set prices cannot be defined explicitly for general values of the number of firms, and therefore the probability of search regret cannot be computed. The cumulative distribution function is defined explicitly only when $N \to \infty$ and $N = 2$ .

A consumer who incurs the search cost

n \in {1, 2}

times incurs a total search cost equal to nc. The detailed timing of the non-sequential search game is as follows. Firms announce prices and consumers choose their search intensity simultaneously. The solution concept is that of Nash equilibrium. Denote θ_n the probability that the representative consumer with positive search costs incurs the search cost

n \in {1, 2}

times. Each firm optimally chooses its price conditional on the price distribution of the rival firm and

{θ_{n}}_{n = 1}^{2}

. Each consumer optimally chooses

{θ_{n}}_{n = 1}^{2}

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) the per firm profits attained. An equilibrium is represented by

{F (p), {θ_{n}}_{n = 1}^{2}}

, which satisfies (a) given

{θ_{n}}_{n = 1}^{2}

π = \bar{π}

\forall p

in the support of F(p), where

\bar{π}

is a constant and

π \leq \bar{π}

\forall p

outside the support of F(p) and (b)

{θ_{n}}_{n = 1}^{2}

is optimal, given that the beliefs of the consumers about the distributions of prices prove correct.

Analysis

My result is presented in Proposition 1.

Proposition 1. Whenever Equation (2) admits a solution $θ_{1} \in (0, 1)$ which satisfies Inequality (1) where

v (1 - \frac{θ_{1}}{2 (1 - θ_{1})} \ln (\frac{2 - θ_{1}}{θ_{1}})) - (\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{c}{v - c}) γ > c

(1)

\begin{array}{l} (\frac{θ_{1} v}{2 (1 - θ_{1})}) (\frac{1}{1 - θ_{1}} \ln (\frac{2 - θ_{1}}{θ_{1}}) - 2) \\ + ((\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{c}{v - c}) - {((\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{2 c}{v - 2 c}))}^{2}) γ = c, \end{array}

(2)

then there exists a dispersed price equilibrium whereby all firms set prices according to the symmetric atomless cumulative distribution function $F (p) = 1 - (\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{v - p}{p})$ with support $[\frac{θ_{1} v}{2 - θ_{1}}, v]$ and θ₁ + θ₂ = 1.

The probability of search regret for the consumers with positive search costs who search once is equal to $1 - F (v - c) = (\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{c}{v - c})$ , and the probability of search regret for the consumers with positive search costs who search twice is ${(1 - F (v - 2 c))}^{2} = {((\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{2 c}{v - 2 c}))}^{2}$ . Namely, a consumer with positive search costs who searches once experiences search regret when the price observed is higher than v–c, and a consumer with positive search costs who searches twice experiences search regret when both prices observed are higher than v–2c. Denote $Φ (θ_{1}; γ, c)$ the left-hand side of Equation (2). Following Fershtman and Fishman (1992, 1994), I categorize the search equilibria according to a stability property. To that end, Figure 1 proves useful.

From Figure 1 it is evident that there exist two solutions to Equation (2); one in the increasing part of

Φ (θ_{1}; γ, c)

that is characterized by a low value of θ₁ and one in the decreasing part of

Φ (θ_{1}; γ, c)

that is characterized by a high value of θ₁. The equilibrium with the low value of θ₁ is stable, whereas the equilibrium with the high value of θ₁ is unstable. Observe that on the left of the equilibrium with the low value of θ₁, θ₁ tends to increase, and on the right of the equilibrium with the low value of θ₁, θ₁ tends to decrease. Whereas on the left of the equilibrium with the high value of θ₁, θ₁ tends to decrease, and on the right of the equilibrium with the high value of θ₁, θ₁ tends to increase.

Table 1:

Simulations – Non-sequential Search Benchmark Parameters: v = 100, λ = 0, $λ \in [0, 1)$ . For the parameter combinations of this Table Equation (2) admits two solutions: the stable $θ_{1}$ and the unstable $θ_{1}$ . The left-hand side (LHS) of Inequality (1) both for the stable and the unstable $θ_{1}$ takes values higher than the search cost.

	Stable $θ_{1}$ (LHS of (1))	Unstable $θ_{1}$ (LHS of (1))
c = 1	0.00493 (98.51130) accepted	0.96905 (3.06306) accepted
c = 2	0.01265 (96.76009) accepted	0.93605 (6.26624) accepted
c = 3	0.02286 (94.78217) accepted	0.90066 (9.63537) accepted
c = 4	0.03582 (92.56050) accepted	0.86244 (13.20541) accepted
c = 5	0.05205 (90.05521) accepted	0.82078 (17.02565) accepted
c = 6	0.07235 (87.19869) accepted	0.77479 (21.16961) accepted
c = 7	0.09810 (83.87624) accepted	0.72305 (25.75612) accepted
c = 8	0.13180 (79.87429) accepted	0.66302 (31.00173) accepted
c = 9	0.17903 (74.70741) accepted	0.58907 (37.39420) accepted
c = 10	0.26081 (66.52678) accepted	0.48017 (46.78419) accepted
c = 20	Not Available	Not Available
c = 30	Not Available	Not Available

Table 2:

Simulations – Non-sequential Search with Search Regret Disutility Parameters: v = 100 , $γ = 10000$ , $λ \in [0, 1)$ . For the parameter combinations of this Table Equation (2) admits two solutions: the stable $θ_{1}$ and the unstable $θ_{1}$ . The left-hand side of Inequality (1) for the stable values of $θ_{1}$ takes values higher than the search cost. The left-hand side of Inequality (1) for the unstable values of $θ_{1}$ takes negative values.

	Stable $θ_{1}$ (LHS of (1))	Unstable $θ_{1}$ (LHS of (1))
c = 1	0.00375 (98.62781) accepted	0.97979 (-2447.40302) rejected
c = 2	0.00691 (97.31704) accepted	0.95918 (-2394.27859) rejected
c = 3	0.00910 (96.10272) accepted	0.93817 (-2340.62699) rejected
c = 4	0.01062 (94.95435) accepted	0.91676 (-2286.44918) rejected
c = 5	0.01169 (93.84857) accepted	0.89495 (-2231.74694) rejected
c = 6	0.01246 (92.77076) accepted	0.87274 (-2176.52290) rejected
c = 7	0.01303 (91.71185) accepted	0.85014 (-2120.78066) rejected
c = 8	0.01345 (90.66601) accepted	0.82717 (-2064.52488) rejected
c = 9	0.01376 (89.62937) accepted	0.80383 (-2007.76139) rejected
c = 10	0.01399 (88.59922) accepted	0.78012 (-1950.49730) rejected
c = 11	0.01424 (78.37495) accepted	0.52755 (-1353.11609) rejected
c = 12	0.01333 (67.66779) accepted	0.27082 (-730.31817) rejected

Table 3:

Simulations – Non-sequential Search with Search Regret Disutility Parameters: $v = 100$ , $γ = 1$ , $λ \in [0, 1)$ . For the parameter combination of this Table Equation (2) admits two solutions: the stable $θ_{1}$ and the unstable $θ_{1}$ . The left-hand side of Inequality (1) both for the stable and the unstable $θ_{1}$ takes values higher than the search cost.

	Stable $θ_{1}$ (LHS of (1))	Unstable $θ_{1}$ (LHS of (1))
c = 1	0.00493 (98.51131) accepted	0.97074 (2.73000) accepted

Tables 1 and 3 demonstrate that when the parameter γ is zero or more generally low, then all values of θ₁ which satisfy Equation (2) also satisfy Inequality (1), that is, the unstable solutions of Equation (2) are accepted as equilibria of the game. On the contrary, Table 2 shows that when the parameter γ is high, the unstable solutions of Equation (2) do not satisfy Inequality (1), and therefore these solutions are not accepted as equilibria of the game.⁵

The economic mechanism behind the uniqueness result is that when γ increases, the unstable solution of Equation (2) increases (see Tables 1 –3). In turn, an increase in the unstable solution of Equation (2) increases both the expected price and the probability of search regret.

As a result, it is most meaningful to focus on the stable solutions of Equation (2).

Figure 1:

Source: The author.

I demonstrate numerically through Tables 1 –3 that an increase in γ decreases the value of the stable solution of Equation (2). It is analytically obvious that a decrease in θ₁ decreases both the probability of search regret and the lower bound of the support. Hence, the consumers with zero search costs, who expect to pay a price equal to the lower bound of the support (see Proof of Proposition 1), are better off, that is, an increase in γ creates a positive externality. It turns out that an increase in γ additionally benefits the consumers with positive search costs. This is shown numerically in Table 4.

Table 4:

Simulations – Non-sequential Search with Search Regret Disutility Parameters: $v = 100$ , c = 1, $λ \in [0, 1)$ . For all values of $γ$ , the Net Payoff is computed for the stable $θ_{1}$ .

Net Payoff per Consumer with Positive Search Costs
$γ = 0$	97.51130
$γ = 1$	97.51131
$γ = 10000$	97.62781

Noisy Sequential Search

The main assumptions of Section 3 continue to hold. The noisy search approach proposed by BJ83 is as follows. The representative consumer with positive search costs by incurring the search cost once receives an unknown number of price quotes; the probability any particular number of price quotes will be observed is known. In the BJ83 original model, by searching once, a consumer with positive search costs observes one price with some probability, two prices with some probability, three prices with some probability, and so on until infinity, that is, a consumer may observe infinite prices with positive probability by searching once. In this paper, I follow Cason and Friedman (2003), and I assume that by searching once, a consumer with positive search costs observes one price with probability $q \in (0, 1)$ and two prices with probability $1 - q$ . As Cason and Friedman put it, samples of size three and larger do not have a penetrating effect on the BJ83 analysis anyway. After the observation of the first sample, the consumers with positive search costs can either purchase at the lowest price of this sample or sample again. Before sampling, the consumers with positive search costs hold a belief about the cumulative distribution function F(p) on the basis of which the firms set prices, and the distribution of the lowest price is $G (p) = q F (p) + (1 - q) (1 - {(1 - F (p))}^{2})$ . The consumers with positive search costs behave according to the reservation price rule: they purchase the product if the lowest price in the current sample is equal to or lower than p^*, and they search again if the lowest price in the current sample is higher than p^*, where p^* ≤ v. An equilibrium is represented by ${F (p), p^{*}}$ , and (a) given q and p^*, F(p) is a Nash equilibrium; and (b) given c, q and F(p), p^* is an optimal reservation price strategy. I focus only on parameters that satisfy Assumption 1.

Assumption 1: c > (\frac{1 - q}{2 - q}) v

The rationale for this restriction is that when $c < (\frac{1 - q}{2 - q}) v$ , the equilibrium reservation price is lower than v–c, and the consumers with positive search costs do not experience probabilistic search regret.⁶

In Appendix B, I show that Assumption 1 leads to probabilistic search regret when γ = 0. In the setting with disutility, generic parameters that do not satisfy Assumption 1 could lead to multiplicity of equilibria in the specific sense that two reservation prices could exist; one reservation price below v – c and one reservation price above v – c. Extensive simulation exercises reveal that for $c < (\frac{1 - q}{2 - q}) v$ and γ > 0 only one reservation price exists, which is lower than v – c.

For convenience in the presentation of my result in the ‘Analysis’ subsection, let A denote the expression

(\frac{q^{2}}{2 (1 - q)}) (\frac{z}{v - c} - 1) + (\frac{q^{2}}{4 (1 - q)}) {(\frac{z}{v - c} - 1)}^{2},

and let B denote the expression

q (1 + (1 - \frac{z}{v - c}) (\frac{q}{2 (1 - q)})) + (1 - q) (1 - {((\frac{z}{v - c} - 1) (\frac{q}{2 (1 - q)}))}^{2})

Analysis

The noisy sequential search equilibrium is characterized in Proposition 2.

Proposition 2. Whenever Equation (3) admits a solution $z^{*} \in (v - c, v)$ and conditions (a)–(d) hold where

z (1 - q) + γ A B = c

(3)

(a) $z (1 - q) + γ A B > c$ for $z > z^{*}$ ,

(b) $z (1 - q) + γ A B > c$ for $z \in [\frac{q z^{*}}{2 - q}, z^{*}),$

(d) $v - z^{*} q - γ (\frac{z^{*}}{v - c} - 1) (\frac{q}{2 (1 - q)}) - c > 0$ ,

then there exists a unique equilibrium. The reservation price of the consumers with positive search costs is $p^{*} = z^{*}$ and all firms set prices according to the symmetric atomless cumulative distribution function $F (p) = 1 - (\frac{q}{2 (1 - q)}) (\frac{p^{*}}{p} - 1)$ with support $[\frac{q p^{*}}{2 - q}, p^{*}]$ . The consumers with positive search costs search once and expect to pay a price equal to $p^{*} q$ .

In Proposition 2, the reservation price belongs to the interval (v–c, v) and the lower bound of the support is lower than v–c (see condition (c)).⁷

The reservation price equals v whenever Equation (3) does not admit a solution in (v – c,v) and conditions $(a' - c')$ hold where $(a' - c')$ $z (1 - q) + γ A B < c$ for $z \in [\frac{q v}{2 - q}, v]$ ; $(b^{'})$ $\frac{q v}{2 - q} < v - c$ ; and $(c^{'})$ $v - v q - γ (\frac{v}{v - 1} - 1) (\frac{q}{2 (1 - q)}) - c > 0$ . I do not emphasize this case, because simulations suggest that the participation constraint, that is, condition , is not satisfied. Consider , for example, the parameters v = 100, q = 0.5, γ = 30 and c = 60.

Thus, probabilistic search regret is established. The first observation is that when γ > 0, the participation constraint for the consumers with positive search costs is not satisfied for all values of the search cost for which it is satisfied when γ = 0 (see Tables 5–6).

Table 5:

Simulations – Noisy Sequential Search with No Search Regret Disutility (Probabilistic Search Regret) Parameters: $v = 100$ , $q = 0.5$ , $γ = 0$ , $λ \in [0, 1)$ . The upper bound of the support is 80, the lower bound of the support is approximately equal to 26.6, and the expression $(\frac{1 - q}{2 - q}) v$ is approximately equal to 33.3. For probabilistic search regret to occur, according to Proposition B.2, the search cost must satisfy $(\frac{1 - q}{2 - q}) v < c < (1 - q) v$ . The Inequality $c < (1 - q) v$ is the participation constraint for the consumers with positive search costs.

Participation Constraint for Consumers with Positive Search Costs

c < 50

Table 6:

Simulations – Noisy Sequential Search with Search Regret Disutility Parameters: $v = 100$ , $q = 0.5$ , $γ = 0$ , $λ \in [0, 1)$ . For the case $γ = 30$ , c = 50: the reservation price is approximately equal to 87.075, the lower bound of the support is approximately equal to 29.025, the probability of search regret is approximately equal to 0.37, and the net payoff of the consumers with positive search costs is approximately equal to -3.66; that is, the conditions (a)-(c) of Proposition 2 are satisfied, and the condition (d) of Proposition 2 is not satisfied. For the case $γ = 10000$ , c = 45: the reservation price is approximately equal to 55.38, the lower bound of the support is approximately equal to 20, the probability of search regret is approximately equal to 0.003456, and the net payoff of the consumers with positive search costs is approximately equal to -7.25; that is, the conditions (a)-(c) of Proposition 2 are satisfied, and the condition (d) of Proposition 2 is not satisfied.

Participation Constraint for Consumers with Positive Search Costs
$γ = 30$ $c = 49$	Not Satisfied
$γ = 10000$ $c = 45$	Not Satisfied

For example, when v = 100, q = 0.5 and γ = 0, there exists a noisy sequential search equilibrium for

c \in (33.3, 50)

(see Proposition B.2, Appendix B). However, when v = 100, q = 0.5 and γ = 30, a noisy sequential search equilibrium does not exist for c = 49. As the disutility increases, that is, when v = 100, q = 0.5 and γ = 10000, a noisy sequential search equilibrium does not exist for c = 45. It can be shown that when v = 100, q = 0.5 and γ = 30, a noisy sequential search equilibrium exists for c = 45. Next, I fix the search cost at a level that satisfies Assumption 1, and I highlight through Table 7 that as γ increases, the reservation price decreases.

Table 7:

Simulations – Noisy Sequential Search Parameters: $v = 100$ , $q = 0.5$ , $c = 40$ , $λ \in [0, 1)$ . For all positive values of $γ$ in this Table, the conditions (a)-(d) in Proposition 2 are satisfied. For $γ = 0$ , the conditions in Proposition B.2 (Appendix B) are satisfied.

	Reservation Price	Net Payoff per Consumer with Positive Search Costs
$γ = 0$	80	20
$γ = 30$	75.8496	18.11279
$γ = 50$	73.938	21.47272
$γ = 80$	71.8154	16.215
$γ = 10000$	60.2369	10.137

As mentioned in the Introduction, the intuition for the reservation price decrease in response to an increase in γ is that the consumers with positive search costs attempt to decrease the probability of search regret. Indeed, when the reservation price decreases, the probability of search regret decreases, the expected prices decrease and the lower bound of the support decreases. Similarly, to the non-sequential search environment, an increase in γ creates a positive externality, as the consumers with zero search costs purchase the product at lower price. In contrast to Table 4, Table 7 shows that an increase in γ does not necessarily benefit the consumers with positive search costs.

Conclusion

I revisited the BJ83 model by introducing search regret disutility. In the presence of disutility, the consumers with positive search costs through their behaviour try to decrease the probability of search regret. This is true both in the non-sequential search approach and in the noisy sequential search approach. In the non-sequential search approach, the consumers increase their search intensity when the disutility increases. In the noisy sequential search approach, the consumers with positive search costs decrease their reservation price when the disutility increases. Furthermore, when the disutility takes very high values, multiplicity of the non-sequential search equilibria disappears. An interesting theme would be the impact of search regret disutility on the social welfare with non-sequential search when new firms enter the industry. Unfortunately, as mentioned in the ‘Non-sequential Search’ section, in the BJ83 equilibrium with consumer heterogeneity, the cumulative distribution function cannot be defined explicitly for general values of the number of firms, and therefore the probability of search regret cannot be computed. Nevertheless, one could examine whether other types of equilibria that do not appear in the original BJ83 model and its extensions exist in the presence of search regret disutility.

Appendix A: Omitted Proofs (Main Body)

Proof of Proposition 1. The proof follows Janssen and Moraga-González (2004, Equation (7) and Proof of Proposition 6). For the purposes of the proof, the number of firms is considered arbitrary, that is, N ≥ 2. Obviously, no firm has an incentive to set a price strictly higher than v; a firm that charges a price above v gets zero demand. A consumer with positive search costs who searches once observes the price of a particular seller with probability $\frac{1}{N}$ , whereas a consumer with positive search costs who searches twice observes the price of a particular seller with probability $\frac{2}{N}$ . When each consumer with positive search costs randomizes between incurring the search cost once and incurring the search cost twice, that is, when θ₁ + θ2 = 1 with θ₁ > 0 and θ2 > 0, then each firm can secure profits equal to $\frac{θ_{1} (1 - λ) v}{N}$ . The expected profit of a firm which charges p (≤ v) when the rival firm sets prices according to the cumulative distribution function F(p) and each consumer with positive costs randomizes between searching once and searching cost twice is

p (\frac{θ_{1} (1 - λ)}{N} + \frac{2 (1 - θ_{1}) (1 - λ)}{N} (1 - F (p)) + λ {(1 - F (p))}^{N - 1}).

This expected profit is analysed as follows. The firm captures the consumers who observe only one price with probability $\frac{1}{N}$ even if it charges the highest price in the market. The firm captures an additional mass equal to $(1 - θ_{1}) (1 - λ)$ , that is, all consumers who observe two prices, with probability $\frac{2}{N}$ if it charges a lower price than a rival firm. In equilibrium, the firms play no atoms. The reason is that if all firms played a price with positive probability, there would be a positive probability of a tie. In case of a tie, the consumers with zero search costs would select a firm at random. Then, one of the rival firms would have a profitable unilateral deviation through a slight decrease in that price that would result in capturing all consumers with zero search costs. The case whereby all firms set a price equal to the marginal cost with positive probability is ruled out, because any firm would have a unilateral incentive to increase the price above marginal cost due to the existence of consumers who observe only one price. Next, the expected profits must be equal to $\frac{θ_{1} (1 - λ) v}{N}$ , that is:

p (\frac{θ_{1} (1 - λ)}{N} + \frac{2 (1 - θ_{1})}{N} (1 - F (p)) + λ {(1 - F (p))}^{N - 1}) = \frac{θ_{1} (1 - λ) v}{N} .

(A.1)

As N increases, the term $λ {(1 - F (p))}^{N - 1}$ approaches zero faster than the other terms. Therefore, when $N \to \infty$ , firms ignore the consumers with zero search costs, and from (A.1) it is easily derived that $F (p) = 1 - (\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{v - p}{p})$ with support , where the lower bound of the support $[\frac{θ_{1} v}{2 - θ_{1}}, v]$ satisfies $F (\underline{p}) = 0$ . In equilibrium, the expected price equals $E (p) = \frac{θ_{1} v}{2 (1 - θ_{1})} \ln (\frac{2 - θ_{1}}{θ_{1}}),$ and the expected minimum price when two prices pi, pj are observed equals $E (\min {p_{i}, p_{j}}) = \frac{θ_{1} v}{1 - θ_{1}} (1 - \frac{θ_{1}}{2 (1 - θ_{1})} \ln (\frac{2 - θ_{1}}{θ_{1}}))$ . Conditional on these prices, each consumer with positive search costs randomizes between searching once and searching twice when searching once delivers the same pay-off as searching twice, that is, when Equation (2) is satisfied, and when searching once delivers a strictly positive pay-off, that is, when Inequality (1) is satisfied. Note that for the consumers with positive search costs, the pay-off from incurring the search cost once equals

v - E (p) - (\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{c}{v - c}) γ - c

and the pay-off from incurring the search cost twice equals

v - E (\min {p_{i}, p_{j}}) - {((\frac{θ_{1}}{2 (1 - θ_{1})}) (\frac{2 c}{v - 2 c}))}^{2} γ - 2 c .

Also, note that in equilibrium, the probability of partial search regret when one price is observed is strictly positive and lower than one. The same holds in equilibrium for the probability of partial search regret when two prices are observed. In equilibrium, the consumers with zero search costs obtain pay-off equal to $v - \frac{θ_{1} v}{2 - θ_{1}}$ , that is, they purchase the product at a price equal to the lower bound of the support. More precisely, a consumer who samples n prices from the distribution F(p) with support $[\frac{θ_{1} v}{2 - θ_{1}}, v]$ and purchases from the lowest price in the sample expects to pay $\int_{\frac{θ_{1} v}{2 - θ_{1}}}^{v} n p {(1 - F (p))}^{n - 1} d F (p)$ . By partial integration, one obtains $\int_{\frac{θ_{1} v}{2 - θ_{1}}}^{v} n p {(1 - F (p))}^{n - 1} d F (p) = \frac{θ_{1} v}{2 - θ_{1}} + \int_{\frac{θ_{1} v}{2 - θ_{1}}}^{v} {(1 - F (p))}^{n} d p$ . It is obvious that when $n \to \infty$ , the term $\int_{\frac{θ_{1} v}{2 - θ_{1}}}^{v} {(1 - F (p))}^{n} d p$ tends to zero. Q.E.D.

Proof of Proposition 2. Given a reservation price $p^{*} \in (v - c, v)$ with $\frac{q p^{*}}{2 - q} < v - c$ , the cumulative distribution function is $F (p) = 1 - (\frac{q}{2 (1 - q)}) (\frac{p^{*} - p}{p})$ with support $[\frac{q p^{*}}{2 - q}, p^{*}]$ (see the arguments in the Proof of Proposition 1). The difference with the setting with γ = 0 (see Appendix B) is the following. Here, if the consumers with positive search costs decide to search once, the lowest price they will sample is lower than v – c with some probability and higher than v – c with the remaining probability. If the lowest price from the first sample is higher than v – c, then the expected benefit from another search is

\int_{0}^{v - c} (z - p + γ) d G (p) + \int_{v - c}^{z} (z - p) d G (p) .

The reasoning is that if the lowest price from another search is lower than v – c, then the representative consumer with positive search costs gains not only from the decrease in the price but also from not incurring the lump sum expenditure γ. The reservation price p^* is the solution z^* to Equation (A.2)

A (\int_{0}^{v - c} (z - p + γ) d G (p) + \int_{v - c}^{z} (z - p) d G (p)) + B (\int_{0}^{z} (z - p) d G (p)) = c

(A.2)

A is described in the main body of the paper and represents the probability $q (1 - F (v - c)) + (1 - q) {(1 - F (v - c))}^{2}$ with p^* = z. In words, A represents the probability that the lowest price from the first sample is higher than v – c. B is described in the main body of the paper too and represents the probability $q F (v - c) + (1 - q) (1 - {(1 - F (v - c))}^{2})$ with p^* = z. In words, B represents the probability that the lowest price from the first sample is lower than v – c. It is obvious that A + B = 1. The left-hand side of Equation (A.2) is equal to the left-hand side of Equation (3). Whenever Equation (3) has a solution $z^{*} \in (v - c, v)$ and conditions (a)–(d) hold, then there exists a unique equilibrium as described in Proposition 2. Q.E.D.

Appendix B: Noisy Sequential Search (BJ83)

Perfect Competition with No Search Regret Disutility (γ = 0)

Assumptions

The assumptions described in section ‘The Model: Main Assumptions’ of the main body of the paper hold, and the set-up is the same as the one described in section ‘Noisy Sequential Search’ of the main body of the paper. The only difference is that γ = 0.

Analysis

First, I show that when the search cost is relatively low, that is, when $c < (\frac{1 - q}{2 - q}) v,$ then the reservation price of the consumers with positive search costs is lower than v – c , and there is no search regret.

Proposition B.1. (No Search Regret) Suppose $c < (\frac{1 - q}{2 - q}) v$ . Then, there exists a unique equilibrium. The reservation price of the consumers with positive search costs is $p^{*} = \frac{c}{1 - q} < v - c$ , and all firms charge prices according to the cumulative distribution function $F (p) = 1 - (\frac{q}{2 (1 - q)}) (\frac{c}{p (1 - q)} - 1)$ with support $[\frac{q c}{(2 - q) (1 - q)}, \frac{c}{1 - q}]$ . The consumers with positive search costs search once and expect to pay a price equal to $\frac{q c}{(1 - q)}$ .

Proof of Proposition B.1. The reservation price p^* is the solution z^* to Equation (B.1)

c = \int_{0}^{z} (z - p) d G (p)

(B.1)

The right-hand side of Equation (B.1) is the expected benefit from search and equals $\int_{0}^{z} G (p) d p$ by partial integration. Given the reservation price p^*, the cumulative distribution function is $F (p) = 1 - (\frac{q}{2 (1 - q)}) (\frac{p^{*} - p}{p})$ with support $[\frac{q p^{*}}{2 - q}, p^{*}]$ (see the arguments in Appendix A, Proof of Proposition 1). Then, the right-hand side of Equation (B.1) equals z(1 – q) with p^* = z, so $p^{*} = z^{*} = \frac{c}{1 - q}$ . By assumption, $c < (\frac{1 - q}{2 - q}) v$ ; this means that $\frac{c}{1 - q} < v - c$ . Provided that the upper bound of the support is lower than v – c, the consumers with positive search costs search once and expect to pay a price equal to $q \int_{\frac{q c}{(2 - q) (1 - q)}}^{\frac{c}{1 - q}} p d F (p) + 2 (1 - q) \int_{\frac{q c}{(2 - q) (1 - q)}}^{\frac{c}{1 - q}} p (1 - F (p)) d F (p) = \frac{q c}{1 - q}$ . The consumers with zero search costs purchase the product at a price equal to the lower bound of the support (see the arguments in Appendix A, Proof of Proposition 1). Q.E.D.

Proposition B.2. (Probabilistic Search Regret) Suppose $(\frac{1 - q}{2 - q}) v < c < (1 - q) v$ . Then, there exists a unique equilibrium. The reservation price of the consumers with positive search costs is $p^{*} = \frac{c}{1 - q} \in (v - c, v)$ , and all firms charge prices according to the cumulative distribution function $F (p) = 1 - (\frac{q}{2 (1 - q)}) (\frac{c}{p (1 - q)} - 1)$ with support $[\frac{q c}{(2 - q) (1 - q)}, \frac{c}{1 - q}]$ . The consumers with positive search costs search once and expect to pay a price equal to $\frac{q c}{1 - q}$ .

Proof of Proposition B.2. The steps of the proof are the same as the steps in the proof of Proposition B.1. The constraint $(\frac{1 - q}{2 - q}) v < c$ means that $\frac{c}{1 - q} > v - c$ , and the constraint c < (1 – q)v means that $\frac{c}{1 - q} < v$ . The constraint c < (1 – q)v is the participation constraint as it guarantees that the consumers with positive search cost obtain a positive net pay-off by searching once. Q.E.D.

Footnotes

Declaration of Conflicting Interests

Funding

The author received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Evangelos Rouskas

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Equilibrium Price Dispersion with Search Regret Disutility

Abstract

Keywords

Introduction

Literature

The Model: Main Assumptions

Non-sequential Search

Analysis

Simulations – Non-sequential Search with Search Regret Disutility Parameters: v = 100 , c = 1, λ ∈ [ 0 , 1 ) . For all values of γ , the Net Payoff is computed for the stable θ 1 .

Noisy Sequential Search

Analysis

Simulations – Noisy Sequential Search Parameters: v = 100 , q = 0.5 , c = 40 , λ ∈ [ 0 , 1 ) . For all positive values of γ in this Table, the conditions (a)-(d) in Proposition 2 are satisfied. For γ = 0 , the conditions in Proposition B.2 (Appendix B) are satisfied.

Conclusion

Appendix A: Omitted Proofs (Main Body)

Appendix B: Noisy Sequential Search (BJ83)

Perfect Competition with No Search Regret Disutility (γ = 0)

Assumptions

Analysis

Footnotes

Declaration of Conflicting Interests

Funding

ORCID iD

References

Simulations – Non-sequential Search with Search Regret Disutility Parameters: $v = 100$ , c = 1, $λ \in [0, 1)$ . For all values of $γ$ , the Net Payoff is computed for the stable $θ_{1}$ .

Simulations – Noisy Sequential Search Parameters: $v = 100$ , $q = 0.5$ , $c = 40$ , $λ \in [0, 1)$ . For all positive values of $γ$ in this Table, the conditions (a)-(d) in Proposition 2 are satisfied. For $γ = 0$ , the conditions in Proposition B.2 (Appendix B) are satisfied.