Skimming through search

Evangelos Rouskas

doi:10.1515/ger-2019-1067

Published by De Gruyter August 27, 2020

Skimming through search

Evangelos Rouskas

From the journal German Economic Review

https://doi.org/10.1515/ger-2019-1067

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Abstract

I examine a two-period duopolistic market for a durable good where firms compete in prices. Consumers are heterogeneous and can be described according to the following characteristics (i) high valuation and high search intensity; (ii) high valuation and low search intensity; (iii) low valuation and high search intensity; and (iv) low valuation and low search intensity. The market exhibits a new version of the so-called Coasian dynamics. The firms engage in intertemporal price discrimination and only consumers with high valuation and low search intensity purchase the product early. This result is based on a property which dictates that the consumers with high valuation and low search intensity are the most impatient. I call this the skimming through search property. When the difference between the high and the low valuation is small, there is positive probability that the prices in the first period are lower than the prices in the second period, so each firm may set a decreasing sequence of prices in a stochastic sense. Furthermore, when the percentage of consumers with high valuation increases, all consumers pay lower prices. This inter-consumer externality resembles the positive externality caused by an increase in market transparency.

Keywords: search; consumer heterogeneity; intertemporal price discrimination; delay

JEL Classification: L13; D43; D83

Acknowledgment

I thank the Editor Peter Egger and an anonymous referee for comments.

Appendix A Replication of Fudenberg and Tirole (1991)

The material presented in this Appendix is based on Fudenberg and Tirole (1991: Chapter 10).

A.1 Assumptions and setup

A monopolistic firm supplies one product at zero cost and on the demand-side there exist nonatomic consumers with mass equal to one and unit demand. A percentage α of consumers, where α∈(0,1), has valuation for the product equal to vH>0, whereas the remaining percentage 1−α of consumers has valuation for the product equal to vL where 0<vL<vH. The monopolist announces the price as a take-it-or-leave-it offer. All agents have an outside option with a payoff equal to zero. In Subsection A.2 there exists only one period, so the announcement of one price (pM) takes place and the consumers accept or reject it. In Subsection A.3, there exist two periods labeled by t=1,2. In each period the monopolist announces a price and the consumers accept or reject it. In the two-period case, the common discount factor is denoted by δ∈(0,1), and (i) a strategy for the monopolist is a sequence of prices (ptM) that are announced in period t, conditional on the rejection of the previous price, where applicable, (ii) a strategy for the consumers is a choice of acceptance or rejection of the price in each period, conditional on the sequence of past and current prices.

A.2 Static analysis

In the one-period version of the model the consumers behave myopically in the sense that as long as the monopolistic price is lower than or equal to their valuation they accept it with probability one. The results of the static analysis are summarized in Theorem A.1.

Theorem A.1 (Generic values of parameters).

The optimal monopolistic price is as follows:

pM=vL, ifα<vLvH(all consumers accept it).
pM=vH, ifα>vLvH(only consumers with valuationvHaccept it).

Proof of Theorem A.1.

The monopolist will never set a price strictly below v2. The reason is that all prices strictly below v2 attract demand equal to one. However, all consumers buy the product even if the monopolist sets a price equal to vL. The monopolist will never set a price strictly above vH, because in such cases the demand and thus the profit is zero, whereas the monopolist can earn positive profit by setting, for example, a price equal to vH. The monopolist will never set a price in the interval (vL,vH). This is because any price in the interval (vL,vH) attracts only consumers with valuation vH, so the monopolist can earn higher profit by setting a price equal to vH. That is, the optimal monopolistic price is either vH or vL. If α<vLvH, the optimal price is vL with probability one (Part [A]). If α>vLvH, the optimal price is v1 with probability one (Part [B]). □

A.3 Dynamic analysis

In contrast to the one-period version of the model, when the monopolist and the consumers interact over two periods, the consumers with valuation vH do not behave myopically any more. These consumers may reject a price in the first period with positive probability even if that price is lower than or equal to their valuation. However, the consumers with valuation vL continue to behave myopically because in case they delay the purchase of the product until the second period, they receive a zero payoff with probability one. The results of the dynamic analysis are summarized in Theorem A.2.

Theorem A.2 (Generic values of parameters).

The optimal monopolistic price sequence is as follows:

p1M=p2M=vL, ifα<vLvH(all consumers acceptp1M).
Ifα>vLvH, the optimal monopolistic price sequence is the one that delivers the highest profit from the two sequences described below:
1. p1M=(1−δ)vH+δvL,p2M=vL(only consumers with valuationvHacceptp1Mand consumers with valuationvLacceptp2M).
2. p1M=p2M=vH(consumers with valuationvHacceptp1Mwith probability1−(1−α)vLα(vH−vL)and rejectp1Mwith the remaining probability, consumers with valuationvHwho have rejectedp1Macceptp2M, consumers with valuationvLaccept neitherp1Mnorp2M).

Proof of Theorem A.2.

First, I consider Part [A]. When α<vLvH and all consumers reject p1M, the monopolist sets p2M=vL. This reasoning follows from the static analysis. When α<vLvH and only consumers with valuation vL reject p1M, again the monopolist sets p2M=v2. Only two price sequences are candidates for the solution and the monopolist will choose the sequence that delivers the highest profit. The first sequence is the following: p1M=(1−δ)vH+δvL, p2M=vL (only consumers with valuation vH accept p1M and consumers with valuation vL accept p2M). This sequence delivers profit equal to α((1−δ)vH+δvL)+δ(1−α)vL. The second sequence is the following: p1M=p2M=vL (all consumers accept p1M). This sequence delivers profit equal to vL. By the assumption α<vLvH, it holds that vL>α((1−δ)vH+δvL)+δ(1−α)vL, so the monopolist chooses the sequence p1M=p2M=vL and all consumers purchase the product in the first period. It is easily understood that any sequence with p1M∈(vL,(1−δ)vH+δvL) and p2M=vL delivers profit strictly lower than vL. Next, I consider Part [B]. When α>vLvH, then should all consumers reject p1M, the monopolist sets p2M=vH, whereas should only consumers with valuation vL reject p1M, the monopolist sets p2M=vL. Again the sequence p1M=(1−δ)vH+δvL, p2M=vL (only consumers with valuation vH accept p1M and consumers with valuation vL accept p2M) is a candidate for the solution and any sequence with p1M∈(vL,(1−δ)vH+δvL) and p2M=vL is ruled out. Any p1M∈((1−δ)vH+δvL,vH] is rejected by consumers with valuation vL, but cannot be rejected with probability one by consumers with valuation vH. Denote x∈(0,1) the probability according to which the consumers with valuation vH accept p1M∈((1−δ)vH+δvL,vH]. The endogenous value of x must make the monopolist indifferent between setting p2M=vL and setting p2M=vH. That is, x is the solution to the equation: (α(1−x)+1−α)vL=vHα(1−x) or x=1−(1−α)vLα(vH−vL). It holds that x does not depend on p1M. Therefore, the monopolist sets p1M=vH. Furthermore, for the consumers with valuation vH to be indifferent between purchasing the product in the first period with probability x at p1M=vH and purchasing the product in the second period, the monopolist sets p2M=vH with probability one. So, the sequence p1M=p2M=vH (consumers with valuation vH accept p1M with probability 1−(1−α)vLα(vH−vL) and reject p1M with the remaining probability, consumers with valuation vH who have rejected p1M accept p2M, consumers with valuation vL accept neither p1M nor p2M) is another candidate for the solution. Overall, one has to compare αxvH+δα(1−x)vH to α((1−δ)vH+δvL)+δ(1−α)vL. It turns out that the result of such a comparison depends on parameters. □

$Figure A.1 Intertemporal Monopolistic Equilibrium Profits as a Function of the Discount Factor. Parameters: vH=40{v_{H}}=40, vL=39{v_{L}}=39, a=0.99a=0.99 (Theorem A.2 applies). The red color corresponds to static profits. The blue color corresponds to intertemporal price discrimination. The green color corresponds to p1M=p2M=vH{p_{1}^{M}}={p_{2}^{M}}={v_{H}}.$

Figure A.1

Intertemporal Monopolistic Equilibrium Profits as a Function of the Discount Factor. Parameters: vH=40, vL=39, a=0.99 (Theorem A.2 applies). The red color corresponds to static profits. The blue color corresponds to intertemporal price discrimination. The green color corresponds to p1M=p2M=vH.

In Figure A.1 the monopolistic equilibrium profit is depicted as a function of the discount factor. It is evident that the dynamic price discrimination profit is decreasing monotonically in the discount factor. When the discount factor takes high values, it is no longer optimal for the monopolist to price discriminate, i. e., intertemporal price discrimination does not take place for all δ∈(0,1). The profit from intertemporal price discrimination is always lower than the static profit. In the limit, as the discount factor tends to one, the dynamic no-discrimination profits are equal to the static profits.

Appendix B Omitted proofs (main body of the paper)

Proof of Lemma 1.

In a candidate intertemporal price discrimination equilibrium, the mass of consumers that delay consumption equals (1−α)(1−λ)+λ. The construction of F2(p) is as follows. Suppose a firm sets a price which is strictly higher than vL. Then, the firm in question serves zero demand. Consider that a firm sets a price p∈(0,vL]. Then, the firm in question serves demand equal to (1−α)(1−λ)2 even if p is the highest price in the market. This means that no firm ever sets a price which is strictly higher than vL and each firm can secure profit equal to (1−α)(1−λ)vL2. The cumulative distribution function F2(p) solves the equation p(1−α)(1−λ)2+(1−F2(p))pλ=(1−α)(1−λ)vL2. The upper bound of the support of the price distribution is vL, and the lower bound satisfies F2(p)=0. The price distribution function has no atoms and the support has no gaps. The expected price in the second period equals (1−α)(1−λ)vL2λln(1−α)(1−λ)+2λ(1−α)(1−λ), and the expected minimum price in the second period equals (1−α)(1−λ)vLλ1−(1−α)(1−λ)2λln(1−α)(1−λ)+2λ(1−α)(1−λ). Then, r1Hu is the solution to the equation vH−r1Hu=δvH−(1−α)(1−λ)vL2λln(1−α)(1−λ)+2λ(1−α)(1−λ), r1Hi is the solution to the equation vH−r1Hi=δvH−(1−α)(1−λ)vLλ1−(1−α)(1−λ)2λln(1−α)(1−λ)+2λ(1−α)(1−λ), r1Lu is the solution to the equation vL−r1Lu=δvL−(1−α)(1−λ)vL2λln(1−α)(1−λ)+2λ(1−α)(1−λ), and r1Li is the solution to vL−r1Li=δvL−(1−α)(1−λ)vLλ1−(1−α)(1−λ)2λln(1−α)(1−λ)+2λ(1−α)(1−λ). By Inequality (1), r1Hu>r1Hi>r1Lu>r1Li. □

Proof of Lemma 2.

The arguments are the same as the ones developed in the Proof of Lemma 1. The only difference is that now by Inequality (2) r1Lu>r1Hi. □

Proof of Proposition 1.

In equilibrium, the intertemporal profits per seller are α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2. Suppose a firm unilaterally sets a price equal to r1Hi in the first period. Then, the said firm obtains intertemporal profits α(1+λ)2r1Hi+δ(1−λ)(1−α)vL2. In detail, in the first period the firm that deviates attracts both the informed consumers with high valuation and half of the uninformed consumers with high valuation (mass: α(1−λ)2+αλ). All consumers with low valuation delay consumption. So the firm that deviates captures profits equal to δ(1−λ)(1−α)vL2 in the second period as the low-valuation uninformed consumers select a firm at random. Inequality (3) ensures that α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2>α(1+λ)2r1Hi+δ(1−λ)(1−α)vL2. Suppose a firm unilaterally sets a price equal to r1Lu in the first period. Then, the said firm obtains intertemporal profits αλ+1−λ2r1Lu+δ(1−λ)(1−α)vL4. In detail, in the first period the firm that deviates attracts the informed consumers with high valuation, half of the uninformed consumers with high valuation and half of the uninformed consumers with low valuation (mass: αλ+α(1−λ)2+(1−α)(1−λ)2). Following the unilateral deviation, the population mass that delays consumption is equal to (1−α)λ+(1−α)(1−λ)2 and each firm can secure profits equal to δ(1−α)(1−λ)vL4 in the second period as the uninformed consumers with low valuation select a firm at random. Inequality (4) ensures that α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2>αλ+1−λ2r1Lu+δ(1−λ)(1−α)vL4. Suppose a firm unilaterally sets a price equal to r1Li in the first period. Then, the said firm obtains intertemporal profits 1+λ2r1Li+δ(1−λ)(1−α)vL4. In detail, in the first period the firm that deviates attracts all the informed consumers, half of the uninformed consumers with high valuation and half of the uninformed consumers with low valuation (mass: λ+α(1−λ)2+(1−α)(1−λ)2). So the consumers who delay consumption have mass (1−α)(1−λ)2, and the second-period per firm profits are equal to δ(1−α)(1−λ)vL4. Inequality (5) ensures that α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2>1+λ2r1Li+δ(1−λ)(1−α)vL4. Suppose a firm unilaterally sets a price strictly higher than r1Hu in the first period. Then, the consumers who delay consumption have total mass λ+α(1−λ)2+(1−α)(1−λ). Such a composition of demand in the second period yields price dispersion; however, the upper bound of the support of the price distribution in the second period can be either vH or vL. When the maximum price ever charged in the second period is vH, each firm can secure profits equal to δα(1−λ)vH4. When the maximum price ever charged in the second period is vL, each firm can secure profits equal to δ(2−α)(1−λ)vL4. If and only if Inequality (6) holds, then δα(1−λ)vH4≥δ(2−α)(1−λ)vL4. This means that when Inequality (6) holds, a firm that unilaterally sets a price strictly higher than r1Hu in the first period obtains intertemporal profits equal to δα(1−λ)vH4. Inequality (7) ensures that α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2>δα(1−λ)vH4. When Inequality (6) does not hold, then a firm that unilaterally sets a price strictly higher than r1Hu in the first period obtains intertemporal profits equal to δ(2−α)(1−λ)vL4. Inequality (8) is equivalent to α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2>δ(2−α)(1−λ)vL4. □

Proof of Proposition 2.

Similarly to Proposition 1, here, in equilibrium, the intertemporal profits per seller are α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2. Suppose a firm unilaterally sets a price equal to r1Li in the first period. In this case, the arguments are exactly the same as the arguments presented in Proposition 1. Suppose a firm unilaterally sets a price strictly higher than r1Hu in the first period. In this case, again the arguments are exactly the same as the arguments presented in Proposition 1. Suppose a firm unilaterally sets a price equal to r1Lu in t=1. Then, the said firm obtains intertemporal profits equal to 1−λ2r1Lu+δ(1−λ)(1−α)vL4. The reason is that all informed consumers delay consumption and half of the uninformed consumers with low valuation delay consumption as well. That is, a firm that deviates unilaterally attracts half of the uninformed consumers in the first period, and captures expected profits equal to δ(1−λ)(1−α)vL4 in the second period. Inequality (9) is equivalent to α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2>1−λ2r1Lu+δ(1−λ)(1−α)vL4. Suppose a firm unilaterally sets a price equal to r1Hi in the first period. Then, the said firm obtains intertemporal profits equal to 1−λ2+αλr1Hi+δ(1−λ)(1−α)vL4. In the first period, all uninformed consumers that observe the price of the firm that deviates purchase the product immediately. Also, the informed consumers with high valuation purchase the product in the first period from the firm that deviates. The consumers who delay consumption have total mass equal to (1−α)λ+(1−α)(1−λ)2. Inequality (10) is equivalent to α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2>1−λ2+αλr1Hi+δ(1−λ)(1−α)vL4. □

Proof of Proposition 3.

Part [A]: To show that all consumers are better off when α increases, it suffices to show that all prices decrease when α increases.

First Step: The expected price in the second period is decreasing in α

In equilibrium, the expected price in t=2 is E2(p)=(1−α)(1−λ)vL2λln(1−α)(1−λ)+2λ(1−α)(1−λ). The first (partial) derivative of E2(p) with respect to α is represented by the expression

−(1−λ)vLln(1−α)(1−λ)+2λ(1−α)(1−λ)2λ−1(1−α)(1−λ)+2λ

This expression is negative if and only if Inequality (B1) holds:

(B1)ln(1−α)(1−λ)+2λ(1−α)(1−λ)>2λ(1−α)(1−λ)+2λ

When α→0+, the left-hand side of Inequality (B1) equals ln1+λ1−λ and the right-hand side of Inequality (B1) equals 2λ1+λ. Inequality (B2) is true:

(B2)ln1+λ1−λ>2λ1+λ

The left-hand side of Inequality (B2) tends to zero when λ→0+ and increases at a rate equal to 2(1−λ)(1+λ) when λ increases. The right-hand side of Inequality (B2) tends to zero when λ→0+ and increases at a rate equal to 2(1+λ)2 when λ increases. As 2(1−λ)(1+λ)>2(1+λ)2, Inequality (B2) is true ∀λ∈(0,1). So when α→0+, Inequality (B1) is true. The left-hand side of Inequality (B1) increases at a rate equal to 2λ(1−α)((1−α)(1−λ)+2λ) when α increases, and the right-hand side of Inequality (B1) increases at a rate equal to 2(1−λ)((1−α)(1−λ)+2λ)2 when α increases. Obviously 2λ(1−α)((1−α)(1−λ)+2λ)>2(1−λ)((1−α)(1−λ)+2λ)2∀α,λ∈(0,1). Consequently, when α increases, ceteris paribus, the expected price in the second period decreases.

Second Step: The prices in the first period are decreasing in α

In equilibrium, both prices in the first period are equal to r1Hu with probability one where r1Hu=vH(1−δ)+δE2(p). As E2(p) decreases when α increases, ceteris paribus, it holds that r1Hu decreases as well when α increases.

Third Step: The expected minimum price in the second period is decreasing in α

In equilibrium, E2(pmin)=(1−α)(1−λ)vLλ1−(1−α)(1−λ)2λln(1−α)(1−λ)+2λ(1−α)(1−λ). The first (partial) derivative of E2(pmin) with respect to α is represented by the expression

−1−λλ(vL−E2(p))+(1−a)(1−λ)2vLλln(1−α)(1−λ)+2λ(1−α)(1−λ)2λ−1(1−α)(1−λ)+2λ

This expression is negative if and only if Inequality (B3) is true:

(B3)ln(1−α)(1−λ)+2λ(1−α)(1−λ)<λ(1−α)(1−λ)1+(1−α)(1−λ)(1−α)(1−λ)+2λ

When α→0+, the left-hand side of Inequality (B3) equals ln1+λ1−λ and the right-hand side of Inequality (B3) equals 2λ(1−λ)(1+λ). Inequality (B4) is true:

(B4)ln1+λ1−λ<2λ(1−λ)(1+λ)

As mentioned above, ln1+λ1−λ tends to zero when λ→0+ and increases at a rate equal to 2(1−λ)(1+λ) when λ increases. The right-hand side of Inequality (B4) tends to zero when λ→0+ and increases at a rate equal to 2(1+λ2)((1−λ)(1+λ))2 when λ increases. As 2(1−λ)(1+λ)<2(1+λ2)((1−λ)(1+λ))2, Inequality (B4) is true ∀λ∈(0,1). That is, I have shown that when α→0+, Inequality (B3) is true. Similarly to the First Step, the left-hand side of Inequality (B3) increases at a rate equal to 2λ(1−α)((1−α)(1−λ)+2λ) when α increases. The right-hand side of Inequality (B3) increases at a rate equal to λ(1−α)2(1−λ)+λ(1−λ)((1−α)(1−λ)+2λ)2 when α increases. As it holds that 2λ(1−α)((1−α)(1−λ)+2λ)<λ(1−α)2(1−λ)+λ(1−λ)((1−α)(1−λ)+2λ)2∀α,λ∈(0,1), the expected minimum price in t=2 decreases when α increases ceteris paribus.

Part [B]: In equilibrium, the intertemporal per firm profits are equal to α(1−λ)2r1Hu+δ(1−λ)(1−α)vL2 where r1Hu=vH(1−δ)+δE2(p). The first (partial) derivative of the intertemporal per firm profits with respect to α is represented by the expression

1−λ2r1Hu−δvL−α(1−λ)vLδln(1−α)(1−λ)+2λ(1−α)(1−λ)2λ−1(1−α)(1−λ)+2λ

This expression is positive (negative) if and only if Inequality (11) holds (the left-hand side of Inequality (11) is strictly lower than the right-hand side of Inequality (11)).

Part [C]: In equilibrium, the social welfare equals α(1−λ)vH+δ((1−α)vL+αλvH). The first (partial) derivative of this expression with respect to α is vH−δvL−λvH(1−δ)>0. □

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Published Online: 2020-08-27

Published in Print: 2021-05-04

Skimming through search

Abstract

Acknowledgment

Appendix A Replication of Fudenberg and Tirole (1991)

A.1 Assumptions and setup

A.2 Static analysis

Theorem A.1 (Generic values of parameters).

Proof of Theorem A.1.

A.3 Dynamic analysis

Theorem A.2 (Generic values of parameters).

Proof of Theorem A.2.

Appendix B Omitted proofs (main body of the paper)

Proof of Lemma 1.

Proof of Lemma 2.

Proof of Proposition 1.

Proof of Proposition 2.

Proof of Proposition 3.

References

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