Online Appendix to A search model of costly product returns by Vaiva Petrikaitė

Transcription

1 Online Appendix to A search model of costly product returns by Vaiva Petrikaitė 27 May A Early returns Suppose that a consumer must return one product before buying another one. This may happen due to short return periods, long shipping or the shortage of funds of the customer. In this instance, a sequentially shopping consumer pays the return cost before the second purchase and has to pay it again in case of repurchasing the first product. In this section, I derive optimal consumer behaviour and a symmetric equilibrium price under this return regime. I call this setting early returns and refer to the setting in the paper as late returns. A. Optimal shopping To make the expressions of consumer surplus simpler, I use a random variable u that is uniformly distributed on the interval [u, ū]. Furthermore, to account for the differences in ηs and prices, I assume that u i = u j µ, where µ >. Suppose that a consumer shops sequentially, the customer has bought product i, observed u i and contemplates buying product j i. After buying product j, the consumer observes u j and keeps product j or returns it and buys product i again. The first event happens if u j u i r, and the second event happens if u j < u i r. As a result, the expected utility gain from buying product j equals ˆ ūj u i r u j du j ˆ ui r u j (u i r du j r u i = ˆ ūj u i r (u j u i r du j 2r. (A. I define x j u i r, such that (A. equals zero. Then, if u i x j r, the consumer keeps product i and does not buy product j. If the inequality is reversed, then the customer returns product i and buys product j. Similarly, one may derive the utility threshold r that a consumer uses to decide whether to buy product i after purchasing product j. Additionally, I observe that x j µ =.

2 Lemma A.. If a consumer is shopping sequentially in the market with early returns, then the customer buys product i first if > x j. Later, if u i < x j r, the customer buys product j, and returns product j to repurchase product i if u j < u i r. Proof. Consumer surplus that a consumer obtains after buying product i first is denoted by ĈS i. If the consumer buys product j first, the expected surplus equals ĈS j. After performing a few changes of integration variables, the difference ĈS i ĈS j equals ˆ ūi µr u i du i ˆ xi r (ˆ ūi u i r ˆ xi r u j r max{u i r,u i µ} u i du i (ˆ ūi ˆ uj r u i u i r ˆ max{r µ,} (u j µ du j (u j r du i r ˆ ui r u i (u i µ r du j r ( 2 (ū i u i µ r du i du j ˆ r µ ˆ ūi µ r ( 2 (ū i u i r du i u j du j du j. (A.2 If µ r, the derivative of (A.2 with respect to µ is positive: µ r u i r µ 2 (ū i u i µ r ū i µ 2 (ū i u i r = 2r >. If µ > r, the derivative of (A.2 with respect to µ is µ r r u i µ ˆ ūi u i µ r (u j µ du j ˆ ui µ r u i (u i r du j r ū i µ 2 (ū i u i r = ˆ ūi ˆ µ r 2 (ū i u i µ 2r (u j µ du j (u i r du j = u i µ r 2 (ū i u i µ 2r 2ū2 i 2 u2 i u i (µ r 2 (µ r2 µ ( µ r (u i r (µ r = Clearly, (A.2 is concave in µ when µ > r. 2r 2 (µ r2. Further, I take if µ r, and know that (A.2 is more than after setting µ =, which makes (A.2 equal to zero. Thus, ĈS i > ĈS j in this case. Consider µ > r. After setting µ = r, (A.2 is positive: 2

3 ˆ ūi µr u i du i µ ( u i ˆ ūi rµ (u j µ du j = = ˆ xi µr µr ˆ xi µr µr u i du i µ ( u i ū i r µ u i du i µ (u i ū i r µ >. If µ = r u i, then (A.2 is again positive: 2 r ( 4r ū 2 i 2ū i ( 3 x 2 i 6 5 = 2 r ( 2 (ū i 2 6 (ū i 5 >, where the inequality follows from the fact that ū i, the expression 2y 2 6y 5 is decreasing in y, and is positive when y =. Hence ĈS i > ĈS j in this instance. If returns are late and a consumer shops sequentially, then there are expected savings on return costs due to terminating shopping after the first purchase and expected utility loss that occurs when the consumer does not try the second product yielding a higher ex post utility. The expected savings on return costs exceed the expected utility loss and sequential shopping is optimal. If returns are early, there are two additional elements that have to be considered. Firstly, the consumer may keep an inferior product after the second purchase because of additional return costs. Secondly, there are additional expected return costs. As a result under early returns, the consumer may choose to buy both products immediately and later return one instead of sequential shopping. Whether a consumer shops sequentially or non-sequentially, depends of the value of µ. Further, I derive the value of µ that I denote by µ which makes the consumer indifferent between sequential and non-sequential shopping. I assume ū i = further on and I take the difference between consumer surplus when the consumer shops sequentially and buys both products immediately. The consumer is indifferent between both shopping modes if ˆ µr u i du i ˆ xi µr max{r µ,} ˆ µ (ˆ µ udu u i r u j du j ˆ µ (ˆ µ ˆ ui r µ ˆ max{r µ,} u i u j du j (u i r du j r du i ( 2 µ r du i ˆ ui µ u i du j du i r = (A.3 If r µ, t he derivative of the LHS of (A.3 with respect to µ is positive: 4 (2 µ2 x 2 i 2 x >. If r < µ, the derivative is positive too: ( 2 (8 µ x 2 i 2 7 /32. As a result, µ that solves (A.3 is unique. 3

4 Figure A.: The value of µ for different r to µ =. Next, I show that for a high return cost µ =. After setting µ =, the LHS of (A.3 simplifies ( 2 96 ( 9 x 2 i 4. The expression is non-negative if ( /9. As a result, for less than this value If µ r, then sequential search is always optimal because after setting µ = r in (A.3, I obtain ( 2 ( x 4 i 8 x 3 i 8 x 2 i 69 /384 >. In this instance a consumer buys product i only. The value of is greater than or equal to zero if r /4. Suppose that the return cost is higher. Then (A.3 becomes ˆ max{r µ,} u i du i ˆ µ udu ˆ µ (ˆ µ u i u j du j ˆ ui µ u i du j du i r =. (A.4 The derivative of the LHS of (A.4 with respect to µ is positive: ( µ 2 4 µ 2r /2 > (r 2 4r 2r /2 = (r 2 /2 >, r µ; ( µ 2 /2 >, µ > r. If µ =, then (A.4 is r r2. The expression is positive when r [/4, ]. When r >, the LHS of (A.4 is always 2 6 positive. As a result for high r, a consumer shops sequentially and buys product i only. If consumers do not buy the second product then ε looses its value and the model resembles Perloff and Salop (985: a customer buys product i if η i p i > η j p j. Then the effect of the return cost on the symmetric equilibrium price vanishes. As a result, I focus on r /4 further on. From the numerical exercise I find that µ is decreasing in r (Figure A.. Further, I return to the main setting with ηs and define x ε i η p r such that if a consumer shops sequentially, buys product i first and finds ε i > x η p r, then the customer terminates shopping by keeping the good. Otherwise the consumer returns product i and continues shopping. Additionally, I use µ and observe the following. Firstly, if a consumer started shopping from product i in case of sequential shopping, then the indifference condition would imply η = p µ. Secondly, if sequential shopping induces a consumer to buy product j first, then 4

5 indifference condition would imply η = µ p. As a result, the consumers whose p µ shop sequentially by buying product i first; the consumers whose η µ p also shop sequentially by buying product j first. The rest of consumers (whose p µ < η < p µ buy both products at once and return a lower utility providing good. A.2 Equilibrium. Further, I denote the symmetric equilibrium price by p and proceed by deriving the pay-off function of firm i that deviates to p p. Additionally, I assume that µ < r. The seller faces four groups of consumers. The first group of consumers draw very high observable match values of product i and very low observable match values of product j. The difference between η i and η j is sufficiently high so that a consumer does not buy from firm j after observing ε i =. The probability of this event gives the demand by the first group of consumers. This demand is denoted by q i. q i = Pr [η i p i x η j p r] = Γ ( p r x The second group of consumers shop sequentially and start shopping at firm i and later may shop at firm j. These consumers have x r < η p µ. Consider a consumer who belongs to the second group. The customer buys product i and does not return it if u i is above the reservation utility of product j, i.e. ε i η i p i x η j p r. This happens with probability ( x η p r. If the utility of product i is lower than the threshold, then the consumer returns it and buys product j. It may happen that after buying product j the consumer decides to repurchase product i. This happens if ε i η i p i r ε j η j p. The conditional probability of this event equals ˆ x η pr (ε η p r dε. By adding the two probabilities and integrating over the relevant interval of η values, I obtain the demand by the second group of consumers, which is denoted by q i. q i = ˆ p r p r x ˆ p µ p r [ [ ˆ x η pr ( x η p r ( x η p r ˆ x η pr η pr ] (ε p η r dε dγ ( η (ε p η r dε ] dγ ( η The third group of consumers buy both products immediately and keep product i if ε i ε j η p. The demand by these consumers is denoted by q a. 5

6 ˆ p ( ˆ η p q a = p η (ε p η dε dγ ( η p µ ˆ µ p (ˆ (ε p η dε η p dγ ( η p The last group of consumers shop sequentially and start shopping at firm j and buy from firm i only if they return product j. This happens if ε j η j p < x η i p i r. After a fourth-group consumer buys product i, it is kept if ε i x. This is because by retuning product i and again buying product j, the consumer cannot get utility higher than x η i p i. Also, the consumer keeps product i and does not repurchase product j if ε i < x and ε i η i p i > ε j η j p j r. The conditional probability that a consumer buys and keeps product i, given that the shopping started at firm j, is ( x p η r ( x ˆ x max{, η p r} (ε p η r dε. By integrating this conditional probability over the relevant interval of values of η values, I obtain the demand by the fourth group of consumers. I denote this demand by q 2i. q 2i = ˆ pr µ p ˆ x pr r p [ ( x p η r ( x [ ( x p η r ( x The pay-off of firm i is denoted by π i and equals ˆ x ˆ x ] (ε p η r dε dγ ( η η p r (ε p η r dε ] dγ ( η π i = p i ( q i q i q a q 2i. In a symmetric equilibrium, firm i sets p i = p that is given by (A p = 45 6 µ 3 9 x 4 24 µ ( x 2 (A.5 4 x 3 6 x 2 2 x If µ > r, then the second integral of q i and the first integral q 2i followed by appropriate adjustments of the bounds of the integrals. The corresponding value of p in this case is 6

7 Figure A.2: The symmetric equilibrium prices p (dashed and p (solid for different values of r p = 96 ( µ 2 ( x 2 2 x 2 µ ( 7 x 4 2 x 2 2 x 4 x 3 x 6. 6 x 5 2 x 4 4 x 3 39 x 2 42 x. (A.6 The sequential stopping rule under early returns resembles a sequential search model with costly recall as in Janssen and Parakhonyak (24. In fact, after assuming that the first search is costless, deviation prices are observable and recall costs equal search costs, I obtain that the pay-off functions of firms are identical to the ones in the market with costly returns. If recall and search costs are equal and consumers shop sequentially in Janssen and Parakhonyak (24, then the the introduction of recall costs has the same qualitative effect on market equilibrium as an increase in search costs in the market with costless recall. As a result I conclude that, given that all consumers shop sequentially, switching from late returns to early returns is qualitatively equivalent to increasing r in the market with late returns, which results is a lower symmetric equilibrium price. If r is small, then there are consumers who shop non-sequentially in the market with early returns. Their demand is less price elastic (as if r =. However, the rest of the demand dominates and p < p for the whole range of r values (Figure A.2. To see how consumer surplus varies with the return cost, I compute the consumers surplus of This can be easily observed by comparing the symmetric equilibrium prices with and without recall costs with horizontally differentiated products that can be found in the supplementary material Janssen and Parakhonyak (24. 7

8 consumers, who buy from firm i. If µ = This surplus equals ĈS i = ˆ xr ˆ ˆ ηi ˆ ηi x r max{,η i x r} ˆ ˆ min{, xrηi } ˆ x η i max{, η r} ( 2 η i dη j dη i 2 p (ˆ (ε η i dε xr η ( ˆ ( x r η (ε η r (ε η r dε When < µ, the surplus equals x ˆ xr η max{ ηr,} (ε η i r dε dη j dη i. (ε η r (ε η i 2r dε dη j dη i ĈS i = ˆ ˆ ηi x r xr ˆ ˆ max{,ηi µ} max{,η i x r} ˆ ˆ ηi max{η i µ,} ˆ ˆ min{,ηi µ} η i ˆ ˆ min{, xrηi } ˆ x min{,η i µ} max{, η r} ( 2 η i dη j dη i 2 p (ˆ ˆ xr η (ε η i dε (ε η r (ε η i 2r dε dη j dη i xr η max{ ηr,} (ˆ ˆ η (ε η i dε (ε η i (ε η dε r dη j dη i η (ˆ (ε η i (ε η dε r dη j dη i η ( ˆ ( x r η (ε η i r dε (ε η r (ε η r dε x dη j dη i. I compute consumer surplus numerically and compare it with the consumer surplus under late returns. The result is in Figure A.3. When the return cost is small, then consumers are better off under early returns, however, if the return cost is high, then consumer surplus is higher under early returns. The total welfare is lower under early returns. B Free returns In this section, I study whether firms have incentives to eliminate the return cost by simultaneously choosing r and p in the main set-up of the paper, i.e. late returns. I abstract from any costs of the sellers that might be related to this elimination and focus on the strategic reasons of their choice. 8

9 Figure A.3: Consumer surplus CS (dashed and ĈS (solid for different values of r B. Optimal shopping I begin with defining an optimal consumer shopping policy for given prices and return costs. To simplify the expressions of consumer surplus, I abstract from the existence of η and define a random variable ν i u i r j, i = {i, j}, j = {, 2}, i j, that is distributed in the interval [ν i, ν i ] according to a continuous differentiable distribution function Ψ i (ν i. Suppose that a consumer buys product i first, inspects the utility u i and considers buying the other product. If the consumer buys product j, then she returns product j when ν i ν j, and sends back product i when ν j > ν i. As a result, the expected utility after buying product j equals ˆ νi ν j dψ j (ν j ν i dψ j (ν j. ν i ν j The consumer is indifferent between keeping product i and buying product j if the following equality is satisfied: ν i r j = ˆ νi After rearranging equation (B.7, I obtain equation (B.8. ν j dψ j (ν j ν i dψ j (ν j. (B.7 ν i ν j (ν j ν i dψ j (ν j = r j (B.8 ν i Equation (B.8 looks similar to the equation that is used to define a stopping rule in a sequential search model (e.g. equation (7 in Weitzman (979. If equation (B.8 represented the sequential search model, then r j would be the search cost to sample product j and the left-hand-side (LHS of the expression would represent the additional gain from searching product j. In the set-up with return costs, the actual interpretation of (B.8 is similar. The return cost of product j is on the right-hand-side (RHS of the equation and the additional utility gain from purchasing product j is on the LHS of (B.8. However, one must observe that, differently from the sequential search model, 9

10 the LHS of (B.8 also incorporates the return costs of both products. I define x j ν i such that (B.8 is satisfied. Then, if ν i < x j, then the expected utility from buying product j is higher than u i and the consumer should buy product j. If ν i x j, then it is better for the consumer to keep product i and forget about product j. Now I go one step back and compute the total expected consumer surplus from shopping when the consumer buys product i first and later may purchase product j. The consumer keeps product i if either ν i x j or ν j ν i < x j. The customer buys and keeps product j if ν i < x j and ν j > ν i. Then the total expected surplus from shopping equals CS i = ( x j ν i (ν i ν j dψ j (ν j r j ( ˆ νi dψ i (ν i ν j dψ j (ν ν i dψ j (ν j dψ i (ν i. ν i ν i ν j (B.9 I observe that the second summand on the RHS of (B.9 is the expected consumer surplus from shopping when a consumer follows the non-sequential shopping framework, i.e. buys both products immediately and then decides which variety to return. Because the first summand is clearly positive, I conclude that the sequential shopping strategy is the optimal one. Instead of starting with product i, the consumer may buy product j first and then, if the utility of product j is poor, buy product i. threshold value that satisfies By using a similar reasoning to the one above, I derive a (ν i dψ i (ν i = r i. Again, if ν j, then it is worthwhile for the consumer to keep product j instead of buying product i. Otherwise, it is optimal for the customer to purchase product i and chose one of the two. If the consumer buys product j first and later may purchase product i, the total expected consumer surplus from shopping equals (ˆ νi CS j = (ν i ν j dg i (ν i r i dψ j (ν j CS. ν j where CS is the total expected consumer surplus when a consumer buys both products immediately (the second summand of (B.9. Lemma B.2. Suppose that > x j. Then if a consumer that shops optimally, buys product i first and later, if ν i < x j, buys product j. Proof. To prove the proposition, I need check the sign of the following expression.

11 ( x j ν i ( Firstly, by using (B.8 I obtain that ˆ ( νi x j x j ˆ νi x j ( ν i ν j (ν j ν i dψ j (ν j r j (ν i ν j dψ i (ν i r i dψ i (ν i dψ j (ν j ( (ν j ν i dψ j (ν j r j dψ i (ν i = x j ν i (ν j ν i dψ j (ν j ν j dψ j (ν j ν i ν i dψ j (ν j x j (ν j x j dψ j (ν j x j x j dψ j (ν j dψ i (ν i = dψ i (ν i. (B. (B. Secondly, I observe that the derivative of (B. with respect to x j is negative ( Ψ j (x j dψ i (ν i <. x j Therefore, because > x j, I obtain that (B. is more than ν i dψ j (ν j dψ j (ν j ν j dψ j (ν j ν i dψ j (ν j dψ j (ν j ( ˆ νi ν j dψ j (ν j dψ i (ν i = ( ˆ νi x ˆ i νi ν i x ˆ i νj ν i dψ j (ν j dψ i (ν i = (ˆ νi (ν j ν i dψ j (ν j dψ i (ν i r i dψ j (ν j. Finally, by using the last expression, I now that (B. is more than ( ( (ˆ νi ( ν j (ν i ν j dψ i (ν i ν j (ν i ν j dψ i (ν i (ν j ν i dψ j (ν j dψ i (ν i (ν i ν j dψ i (ν i ν j dψ j (ν j dψ j (ν j (ˆ νi ( ν j r i dψ j (ν j dψ j (ν j r i dψ j (ν j = (ν j ν i dψ j (ν dψ i (ν i = (ν i ν j dψ i (ν dψ j (ν i =.

12 To obtain the corresponding thresholds of ε, I rewrite (B.8 in terms of ε, p and r and obtain equation (B.2 ˆ ε i p i p j r j r i (ε j (ε i p i p j r j r i dε j = r j (B.2 I define x j = ε i p i p j r j r i such that (B.2 is satisfied. Then the reservation utility of product j is x j p j r j r i. If a consumer buys product i and observes that ε i p i x j p j r j r i, then product j is never bought. If the inequality is reversed, the customer buys product j and keeps the product that provides with the higher utility. The value x j that is used in the proposition on optimal shopping does not contain r i, because this return cost is in ν j. Therefore, a consumer compares x j p j r j with p i r i before deciding where to go shopping first. If x j p j r j > p i r i, then the buyer buys from firm j first, and buys product i first if the inequality is reversed. B.2 Firm s choices Next, I proceed by considering what a return cost and a prices would be in a symmetric equilibrium. To simplify the expressions for pay-off functions, I define y = y j y i, where i j and y = {p, x, r, η}. A consumer starts shopping from firm i if η p x r, but buys from firm j first if the inequality is reversed. By following a similar reasoning to the one in the paper, I obtain that the corresponding values of q i, q i and q 2i are q i = Γ ( p r x j, and q i = ˆ p x r p r x j [ ˆ xj η p r ] (x j η p r (ε p η r dε dγ ( η q 2i = ˆ p r p x r [ ( p η r ( ˆ xi max{, η r p} ] (ε p η r dε dγ ( η. While raising its return cost, firm i considers two opposite effects of r i on its pay-off. First, there is a positive effect. Namely, a higher return cost discourages consumers from returning product i once it has been bought. However, consumers are rational, and they understand that due to high r i it will be more difficult to dispose of product i if its value is low. Therefore, fewer customers 2

13 start their shopping from firm i. Moreover, if ri increases, then the reservation value xi ηi ri pi decreases. Thus, fewer consumers buy from firm i after shopping at firm j. The last two changes constitute a negative effect of raising the return cost. If the return cost is low initially, then the positive effect of ri is stronger. Therefore, firm i does not set ri = in equilibrium. This result is stated in Proposition 4. By taking the first-order conditions of πi with respect to ri and pi and afterwards setting ri = rj = r and pi = pj = p, one can get the system of equations that have an interior solution, i.e. < r < /2 and the pair {p, r } is unique. Furthermore, it is possible to prove that the pay-off function is locally concave at the candidate equilibrium point. However, the proof of global concavity remains complicated. I rely on numerical simulation results further. With uniformly distributed η and ε, the corresponding values of r and p are.3 and.696. The plot of the pay-off of firm i suggests that the pay-off is not quasi-concave (Figure B.4, although the symmetric equilibrium point seems to give the global maximum. Proof of proposition 4. I prove this proposition by showing that the demand of firm i is increasing in ri when pi = pj and xi = xj. This renders a positive returning cost profitable. The derivative of the demand of firm i with respect to ri equals ˆ xi p r ˆ p x r qi [ η p r ] dγ ( η [ η p r ] dγ ( η = ri p r p r xj γ ( p x r xi ( xj If the seller sets pi = pj and xi = xj, then at the point ri = (xi =, the derivative is lim xi,xj qi ri ˆ [ 2 η ] dγ ( η = = p = (a >. 6 (b Figure B.4: The pay-off of firm i when a firm chooses ri and pi, η is distributed uniformly. 3

14 References Janssen, M. and A. Parakhonyak (24. Consumer search markets with costly revisits. Economic Theory (55, Perloff, J. M. and S. C. Salop (985. Equilibrium with product differentiation. Review of Economic Studies 52 (, 7 2. Weitzman, M. L. (979. Optimal search for the best alternative. Econometrica 47 (3,