Online Appendix to A search model of costly product returns by Vaiva Petrikaitė
|
|
- Avice Adams
- 5 years ago
- Views:
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,
Consumer Obfuscation by a Multiproduct Firm
Consumer Obfuscation by a Multiproduct Firm Vaiva Petrikaitė 2014 July Abstract I show that a multiproduct firm has incentives to obfuscate its varieties by using search costs to screen out its customers
More informationDeceptive Advertising with Rational Buyers
Deceptive Advertising with Rational Buyers September 6, 016 ONLINE APPENDIX In this Appendix we present in full additional results and extensions which are only mentioned in the paper. In the exposition
More informationDesign Patent Damages under Sequential Innovation
Design Patent Damages under Sequential Innovation Yongmin Chen and David Sappington University of Colorado and University of Florida February 2016 1 / 32 1. Introduction Patent policy: patent protection
More informationInformational Complementarity
Informational Complementarity Very Preliminary. Do Not Circulate. T. Tony Ke MIT kete@mit.edu Song Lin Hong Kong University of Science and Technology mksonglin@ust.hk January 2018 Informational Complementarity
More informationDesigning Optimal Pre-Announced Markdowns in the Presence of Rational Customers with Multi-unit Demands - Online Appendix
087/msom070057 Designing Optimal Pre-Announced Markdowns in the Presence of Rational Customers with Multi-unit Demands - Online Appendix Wedad Elmaghraby Altan Gülcü Pınar Keskinocak RH mith chool of Business,
More informationPrices and Heterogeneous Search Costs
Supplementary Appendix to Prices and Heterogeneous Search Costs José Luis Moraga-González Zsolt Sándor Matthijs R. Wildenbeest June 216 Introduction In this supplementary appendix we present two extensions
More informationThe Impact of Advertising on Media Bias. Web Appendix
1 The Impact of Advertising on Media Bias Esther Gal-Or, Tansev Geylani, Tuba Pinar Yildirim Web Appendix DERIVATIONS OF EQUATIONS 16-17 AND PROOF OF LEMMA 1 (i) Single-Homing: Second stage prices are
More informationCournot and Bertrand Competition in a Differentiated Duopoly with Endogenous Technology Adoption *
ANNALS OF ECONOMICS AND FINANCE 16-1, 231 253 (2015) Cournot and Bertrand Competition in a Differentiated Duopoly with Endogenous Technology Adoption * Hongkun Ma School of Economics, Shandong University,
More informationEconomics 2010c: Lectures 9-10 Bellman Equation in Continuous Time
Economics 2010c: Lectures 9-10 Bellman Equation in Continuous Time David Laibson 9/30/2014 Outline Lectures 9-10: 9.1 Continuous-time Bellman Equation 9.2 Application: Merton s Problem 9.3 Application:
More informationCredence Goods and Vertical Product Differentiation: The Impact of Labeling Policies* Ian Sheldon (Ohio State University)
Credence Goods and Vertical Product Differentiation: The Impact of Labeling Policies* Ian Sheldon (Ohio State University) Seminar: North Dakota State University, Fargo, ND, May, 6 * Draws on Roe and Sheldon
More informationAdvanced Microeconomics
Advanced Microeconomics Leonardo Felli EC441: Room D.106, Z.332, D.109 Lecture 8 bis: 24 November 2004 Monopoly Consider now the pricing behavior of a profit maximizing monopolist: a firm that is the only
More informationAdvanced Microeconomic Analysis, Lecture 6
Advanced Microeconomic Analysis, Lecture 6 Prof. Ronaldo CARPIO April 10, 017 Administrative Stuff Homework # is due at the end of class. I will post the solutions on the website later today. The midterm
More informationDepartment of Agricultural Economics. PhD Qualifier Examination. May 2009
Department of Agricultural Economics PhD Qualifier Examination May 009 Instructions: The exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationVertical Product Differentiation and Credence Goods: Mandatory Labeling and Gains from International Integration
Vertical Product Differentiation and Credence Goods: Mandatory Labeling and Gains from International Integration Ian Sheldon and Brian Roe (The Ohio State University Quality Promotion through Eco-Labeling:
More informationIndustrial Organization Lecture 7: Product Differentiation
Industrial Organization Lecture 7: Product Differentiation Nicolas Schutz Nicolas Schutz Product Differentiation 1 / 57 Introduction We now finally drop the assumption that firms offer homogeneous products.
More informationNumerical illustration
A umerical illustration Inverse demand is P q, t = a 0 a 1 e λ 2t bq, states of the world are distributed according to f t = λ 1 e λ 1t, and rationing is anticipated and proportional. a 0, a 1, λ = λ 1
More informationCompetition Policy - Spring 2005 Monopolization practices I
Prepared with SEVI S LIDES Competition Policy - Spring 2005 Monopolization practices I Antonio Cabrales & Massimo Motta May 25, 2005 Summary Some definitions Efficiency reasons for tying Tying as a price
More informationA technical appendix for multihoming and compatibility
A technical appendix for multihoming and compatibility Toker Doganoglu and Julian Wright July 19, 2005 We would like to thank two anonymous referees and our editor, Simon Anderson, for their very helpful
More informationAnswer Key: Problem Set 1
Answer Key: Problem Set 1 Econ 409 018 Fall Question 1 a The profit function (revenue minus total cost) is π(q) = P (q)q cq The first order condition with respect to (henceforth wrt) q is P (q )q + P (q
More informationData Abundance and Asset Price Informativeness. On-Line Appendix
Data Abundance and Asset Price Informativeness On-Line Appendix Jérôme Dugast Thierry Foucault August 30, 07 This note is the on-line appendix for Data Abundance and Asset Price Informativeness. It contains
More informations<1/4 p i = 2 3 s s [1/4, 9/16[ p i = p j + 2 s p j 1 if p j < p j p i = 2 3 s if p j p j p i = p j if p j < p j s 9/16
s
More informationOn Hotelling s Stability in Competition
On Hotelling s Stability in Competition Claude d Aspremont, Jean Jaskold Gabszewicz and Jacques-François Thisse Manuscript received March, 1978; revision received June, 1978 Abstract The purpose of this
More informationOnline Supplementary Appendix B
Online Supplementary Appendix B Uniqueness of the Solution of Lemma and the Properties of λ ( K) We prove the uniqueness y the following steps: () (A8) uniquely determines q as a function of λ () (A) uniquely
More informationSequential Search Auctions with a Deadline
Sequential Search Auctions with a Deadline Joosung Lee Daniel Z. Li University of Edinburgh Durham University January, 2018 1 / 48 A Motivational Example A puzzling observation in mergers and acquisitions
More informationSequential mergers with differing differentiation levels
Sequential mergers with differing differentiation levels March 31, 2008 Discussion Paper No.08-03 Takeshi Ebina and Daisuke Shimizu Sequential mergers with differing differentiation levels Takeshi Ebina
More informationSecond Welfare Theorem
Second Welfare Theorem Econ 2100 Fall 2015 Lecture 18, November 2 Outline 1 Second Welfare Theorem From Last Class We want to state a prove a theorem that says that any Pareto optimal allocation is (part
More informationMathematical Foundations -1- Supporting hyperplanes. SUPPORTING HYPERPLANES Key Ideas: Bounding hyperplane for a convex set, supporting hyperplane
Mathematical Foundations -1- Supporting hyperplanes SUPPORTING HYPERPLANES Key Ideas: Bounding hyperplane for a convex set, supporting hyperplane Supporting Prices 2 Production efficient plans and transfer
More informationHotelling's Location Model with Quality Choice in Mixed Duopoly. Abstract
Hotelling's Location Model with Quality Choice in Mixed Duopoly Yasuo Sanjo Graduate School of Economics, Nagoya University Abstract We investigate a mixed duopoly market by introducing quality choice
More informationCompetitive sequential search equilibrium
Competitive sequential search equilibrium José L. Moraga-González Makoto Watanabe April 20, 2016 Abstract We present a tractable model of a competitive equilibrium where buyers engage in costly sequential
More informationFirms and returns to scale -1- Firms and returns to scale
Firms and returns to scale -1- Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Constant returns to scale 19 C. The CRS economy 25 D. pplication to trade 47 E. Decreasing
More informationCompetitive Advertising and Pricing
Competitive Advertising and Pricing Raphael Boleslavsky Ilwoo Hwang Kyungmin (Teddy) Kim University of Miami November 2018 Raphael Boleslavsky and Ilwoo Hwang This Paper Bertrand competition for differentiated
More informationBresnahan, JIE 87: Competition and Collusion in the American Automobile Industry: 1955 Price War
Bresnahan, JIE 87: Competition and Collusion in the American Automobile Industry: 1955 Price War Spring 009 Main question: In 1955 quantities of autos sold were higher while prices were lower, relative
More informationMechanism Design: Dominant Strategies
May 20, 2014 Some Motivation Previously we considered the problem of matching workers with firms We considered some different institutions for tackling the incentive problem arising from asymmetric information
More informationPrice and Capacity Competition
Price and Capacity Competition Daron Acemoglu, Kostas Bimpikis, and Asuman Ozdaglar October 9, 2007 Abstract We study the efficiency of oligopoly equilibria in a model where firms compete over capacities
More informationOnline Appendix. (S,S) for v v1. (F,F) for v1 S < v
Article submitted to Production & Operations Management 1 Online Appendix Appendix F: When the Retailers Can Decide Whether to Adopt QR Here we describe what happens when the retailers can simultaneously
More informationFigure T1: Consumer Segments with No Adverse Selection. Now, the discounted utility, V, of a segment 1 consumer is: Segment 1 (Buy New)
Online Technical Companion to Accompany Trade-ins in Durable Goods Markets: Theory and Evidence This appendix is divided into six main sections which are ordered in a sequence corresponding to their appearance
More information1 Differentiated Products: Motivation
1 Differentiated Products: Motivation Let us generalise the problem of differentiated products. Let there now be N firms producing one differentiated product each. If we start with the usual demand function
More informationMathematical Appendix. Ramsey Pricing
Mathematical Appendix Ramsey Pricing PROOF OF THEOREM : I maximize social welfare V subject to π > K. The Lagrangian is V + κπ K the associated first-order conditions are that for each I + κ P I C I cn
More informationEstimating Single-Agent Dynamic Models
Estimating Single-Agent Dynamic Models Paul T. Scott New York University Empirical IO Course Fall 2016 1 / 34 Introduction Why dynamic estimation? External validity Famous example: Hendel and Nevo s (2006)
More informationTechnical Appendix to "Sequential Exporting"
Not for publication Technical ppendix to "Sequential Exporting" acundo lbornoz University of irmingham Héctor. Calvo Pardo University of Southampton Gregory Corcos NHH Emanuel Ornelas London School of
More informationECON4510 Finance Theory Lecture 2
ECON4510 Finance Theory Lecture 2 Diderik Lund Department of Economics University of Oslo 26 August 2013 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 2 26 August 2013 1 / 31 Risk aversion and
More informationInert Consumers in Markets with Switching Costs and Price Discrimination
Inert Consumers in Markets with Switching Costs Price Discrimination Marielle C. Non March 15, 2011 Abstract This paper analyzes an infinite-period oligopoly model where consumers incur costs when switching
More informationOptimal Objective Function
Optimal Objective Function Junichi Haraguchi Taku Masuda June 5 017 PRELIMINARY. ANY COMMENTS APPRECIATED. 1 Introduction In a framework of industrial organization we basically assume firms objective is
More informationEmission Quota versus Emission Tax in a Mixed Duopoly with Foreign Ownership
Emission Quota versus Emission Tax in a Mixed Duopoly with Foreign Ownership Kazuhiko Kato and Leonard F.S. Wang December 29, 2012 Abstract The paper compares an emission tax and an emission quota in a
More informationFirms and returns to scale -1- John Riley
Firms and returns to scale -1- John Riley Firms and returns to scale. Increasing returns to scale and monopoly pricing 2. Natural monopoly 1 C. Constant returns to scale 21 D. The CRS economy 26 E. pplication
More informationOn the Unique D1 Equilibrium in the Stackelberg Model with Asymmetric Information Janssen, M.C.W.; Maasland, E.
Tilburg University On the Unique D1 Equilibrium in the Stackelberg Model with Asymmetric Information Janssen, M.C.W.; Maasland, E. Publication date: 1997 Link to publication General rights Copyright and
More informationEntry and Welfare in Search Markets
Entry and Welfare in Search Markets Yongmin Chen and Tianle Zhang May 2014 Abstract. The effects of entry on consumer and total welfare are studied in a model of consumer search. Potential entrants differ
More informationOligopoly Theory. This might be revision in parts, but (if so) it is good stu to be reminded of...
This might be revision in parts, but (if so) it is good stu to be reminded of... John Asker Econ 170 Industrial Organization January 23, 2017 1 / 1 We will cover the following topics: with Sequential Moves
More informationGrowing competition in electricity industry and the power source structure
Growing competition in electricity industry and the power source structure Hiroaki Ino Institute of Intellectual Property and Toshihiro Matsumura Institute of Social Science, University of Tokyo [Preliminary
More informationLecture 1: Labour Economics and Wage-Setting Theory
ecture 1: abour Economics and Wage-Setting Theory Spring 2015 ars Calmfors iterature: Chapter 1 Cahuc-Zylberberg (pp 4-19, 28-29, 35-55) 1 The choice between consumption and leisure U = U(C,) C = consumption
More informationSF2972 Game Theory Exam with Solutions March 15, 2013
SF2972 Game Theory Exam with s March 5, 203 Part A Classical Game Theory Jörgen Weibull and Mark Voorneveld. (a) What are N, S and u in the definition of a finite normal-form (or, equivalently, strategic-form)
More informationMonopoly Regulation in the Presence of Consumer Demand-Reduction
Monopoly Regulation in the Presence of Consumer Demand-Reduction Susumu Sato July 9, 2018 I study a monopoly regulation in the setting where consumers can engage in demand-reducing investments. I first
More informationA Note on Cost Reducing Alliances in Vertically Differentiated Oligopoly. Abstract
A Note on Cost Reducing Alliances in Vertically Differentiated Oligopoly Frédéric DEROÏAN FORUM Abstract In a vertically differentiated oligopoly, firms raise cost reducing alliances before competing with
More informationIndustrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October
Industrial Organization, Fall 2011: Midterm Exam Solutions and Comments Date: Wednesday October 23 2011 1 Scores The exam was long. I know this. Final grades will definitely be curved. Here is a rough
More informationDepartment of Economics, University of California, Davis Ecn 200C Micro Theory Professor Giacomo Bonanno ANSWERS TO PRACTICE PROBLEMS 18
Department of Economics, University of California, Davis Ecn 00C Micro Theory Professor Giacomo Bonanno ANSWERS TO PRACTICE PROBEMS 8. If price is Number of cars offered for sale Average quality of cars
More informationDifferentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries
Differentiable Welfare Theorems Existence of a Competitive Equilibrium: Preliminaries Econ 2100 Fall 2017 Lecture 19, November 7 Outline 1 Welfare Theorems in the differentiable case. 2 Aggregate excess
More informationMechanism Design: Bayesian Incentive Compatibility
May 30, 2013 Setup X : finite set of public alternatives X = {x 1,..., x K } Θ i : the set of possible types for player i, F i is the marginal distribution of θ i. We assume types are independently distributed.
More informationEstimating Single-Agent Dynamic Models
Estimating Single-Agent Dynamic Models Paul T. Scott Empirical IO Fall, 2013 1 / 49 Why are dynamics important? The motivation for using dynamics is usually external validity: we want to simulate counterfactuals
More informationStudy Unit 3 : Linear algebra
1 Study Unit 3 : Linear algebra Chapter 3 : Sections 3.1, 3.2.1, 3.2.5, 3.3 Study guide C.2, C.3 and C.4 Chapter 9 : Section 9.1 1. Two equations in two unknowns Algebraically Method 1: Elimination Step
More informationWireless Network Pricing Chapter 6: Oligopoly Pricing
Wireless Network Pricing Chapter 6: Oligopoly Pricing Jianwei Huang & Lin Gao Network Communications and Economics Lab (NCEL) Information Engineering Department The Chinese University of Hong Kong Huang
More informationLecture 6. Xavier Gabaix. March 11, 2004
14.127 Lecture 6 Xavier Gabaix March 11, 2004 0.0.1 Shrouded attributes. A continuation Rational guys U i = q p + max (V p, V e) + σε i = q p + V min (p, e) + σε i = U i + σε i Rational demand for good
More informationMicroeconomic Theory -1- Introduction
Microeconomic Theory -- Introduction. Introduction. Profit maximizing firm with monopoly power 6 3. General results on maximizing with two variables 8 4. Model of a private ownership economy 5. Consumer
More informationHotelling s Beach with Linear and Quadratic Transportation Costs: Existence of Pure Strategy Equilibria
appearing in Australian Economic Papers, vol. 46(1) Hotelling s Beach with Linear and Quadratic Transportation Costs: Existence of Pure Strategy Equilibria Alain Egli University of Bern Abstract In Hotelling
More informationThe ambiguous impact of contracts on competition in the electricity market Yves Smeers
The ambiguous impact of contracts on competition in the electricity market Yves Smeers joint work with Frederic Murphy Climate Policy and Long Term Decisions-Investment and R&D, Bocconi University, Milan,
More informationA Note of Caution on Using Hotelling Models in Platform Markets
A Note of Caution on Using Hotelling Models in Platform Markets Thomas D. Jeitschko Soo Jin Kim Aleksandr Yankelevich April 12, 2018 Abstract We study a Hotelling framework in which customers first pay
More informationFree and Second-best Entry in Oligopolies with Network
Free and Second-best Entry in Oligopolies with Network Effects Adriana Gama Mario Samano September 7, 218 Abstract We establish an important difference between Cournot oligopolies with and without positive
More informationSealed-bid first-price auctions with an unknown number of bidders
Sealed-bid first-price auctions with an unknown number of bidders Erik Ekström Department of Mathematics, Uppsala University Carl Lindberg The Second Swedish National Pension Fund e-mail: ekstrom@math.uu.se,
More informationOptimal Monopoly Mechanisms with Demand. Uncertainty. 1 Introduction. James Peck and Jeevant Rampal. December 27, 2017
Optimal Monopoly Mechanisms with Demand Uncertainty James Peck and Jeevant Rampal December 27, 2017 Abstract This paper analyzes a monopoly rm's prot maximizing mechanism in the following context. There
More informationNBER WORKING PAPER SERIES PRICE AND CAPACITY COMPETITION. Daron Acemoglu Kostas Bimpikis Asuman Ozdaglar
NBER WORKING PAPER SERIES PRICE AND CAPACITY COMPETITION Daron Acemoglu Kostas Bimpikis Asuman Ozdaglar Working Paper 12804 http://www.nber.org/papers/w12804 NATIONAL BUREAU OF ECONOMIC RESEARCH 1050 Massachusetts
More informationON HOTELLING S COMPETITION WITH GENERAL PURPOSE PRODUCTS 1
ON HOTELLING S COMPETITION WITH GENERAL PURPOSE PRODUCTS 1 CRISTIÁN TRONCOSO VALVERDE AND JACQUES ROBERT 3 Abstract. This paper extends the traditional Hotelling s model of spatial competition by allowing
More informationMonetary Economics: Solutions Problem Set 1
Monetary Economics: Solutions Problem Set 1 December 14, 2006 Exercise 1 A Households Households maximise their intertemporal utility function by optimally choosing consumption, savings, and the mix of
More informationAppendix B for The Evolution of Strategic Sophistication (Intended for Online Publication)
Appendix B for The Evolution of Strategic Sophistication (Intended for Online Publication) Nikolaus Robalino and Arthur Robson Appendix B: Proof of Theorem 2 This appendix contains the proof of Theorem
More informationEconS 501 Final Exam - December 10th, 2018
EconS 501 Final Exam - December 10th, 018 Show all your work clearly and make sure you justify all your answers. NAME 1. Consider the market for smart pencil in which only one firm (Superapiz) enjoys a
More informationProduct Variety, Price Elasticity of Demand and Fixed Cost in Spatial Models
Product Variety, Price Elasticity of Demand and Fixed Cost in Spatial Models Yiquan Gu 1,2, Tobias Wenzel 3, 1 Technische Universität Dortmund 2 Ruhr Graduate School in Economics 3 Universität Erlangen-Nürnberg
More informationMarket Structure and Productivity: A Concrete Example. Chad Syverson
Market Structure and Productivity: A Concrete Example. Chad Syverson 2004 Hotelling s Circular City Consumers are located uniformly with density D along a unit circumference circular city. Consumer buys
More informationElectronic Companion to Tax-Effective Supply Chain Decisions under China s Export-Oriented Tax Policies
Electronic Companion to Tax-Effective Supply Chain Decisions under China s Export-Oriented Tax Policies Optimality Equations of EI Strategy In this part, we derive the optimality equations for the Export-Import
More informationMechanism Design II. Terence Johnson. University of Notre Dame. Terence Johnson (ND) Mechanism Design II 1 / 30
Mechanism Design II Terence Johnson University of Notre Dame Terence Johnson (ND) Mechanism Design II 1 / 30 Mechanism Design Recall: game theory takes the players/actions/payoffs as given, and makes predictions
More informationMarket Equilibrium and the Core
Market Equilibrium and the Core Ram Singh Lecture 3-4 September 22/25, 2017 Ram Singh (DSE) Market Equilibrium September 22/25, 2017 1 / 19 Market Exchange: Basics Let us introduce price in our pure exchange
More information4. Partial Equilibrium under Imperfect Competition
4. Partial Equilibrium under Imperfect Competition Partial equilibrium studies the existence of equilibrium in the market of a given commodity and analyzes its properties. Prices in other markets as well
More informationGame Theory and Algorithms Lecture 2: Nash Equilibria and Examples
Game Theory and Algorithms Lecture 2: Nash Equilibria and Examples February 24, 2011 Summary: We introduce the Nash Equilibrium: an outcome (action profile) which is stable in the sense that no player
More informationEfficient Random Assignment with Cardinal and Ordinal Preferences: Online Appendix
Efficient Random Assignment with Cardinal and Ordinal Preferences: Online Appendix James C. D. Fisher December 11, 2018 1 1 Introduction This document collects several results, which supplement those in
More informationAppendix (For Online Publication) Community Development by Public Wealth Accumulation
March 219 Appendix (For Online Publication) to Community Development by Public Wealth Accumulation Levon Barseghyan Department of Economics Cornell University Ithaca NY 14853 lb247@cornell.edu Stephen
More informationMulti-object auctions (and matching with money)
(and matching with money) Introduction Many auctions have to assign multiple heterogenous objects among a group of heterogenous buyers Examples: Electricity auctions (HS C 18:00), auctions of government
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics July 26, 2013 Instructions The exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationOn production costs in vertical differentiation models
On production costs in vertical differentiation models Dorothée Brécard To cite this version: Dorothée Brécard. On production costs in vertical differentiation models. 2009. HAL Id: hal-00421171
More informationDurable goods monopolist
Durable goods monopolist Coase conjecture: A monopolist selling durable good has no monopoly power. Reason: A P 1 P 2 B MC MC D MR Q 1 Q 2 C Q Although Q 1 is optimal output of the monopolist, it faces
More informationin Search Advertising
Eects of the Presence of Organic Listing in Search Advertising Lizhen Xu Jianqing Chen Andrew Whinston Web Appendix: The Case of Multiple Competing Firms In this section, we extend the analysis from duopolistic
More informationwhere u is the decision-maker s payoff function over her actions and S is the set of her feasible actions.
Seminars on Mathematics for Economics and Finance Topic 3: Optimization - interior optima 1 Session: 11-12 Aug 2015 (Thu/Fri) 10:00am 1:00pm I. Optimization: introduction Decision-makers (e.g. consumers,
More informationRalph s Strategic Disclosure 1
Ralph s Strategic Disclosure Ralph manages a firm that operates in a duopoly Both Ralph s (privatevalue) production cost and (common-value) inverse demand are uncertain Ralph s (constant marginal) production
More informationTechnical Appendix for: Complementary Goods: Creating, Capturing and Competing for Value
Technical Appendix for: Complementary Goods: Creating, Capturing and Competing for Value February, 203 A Simultaneous Quality Decisions In the non-integrated case without royalty fees, the analysis closely
More informationSome Notes on Adverse Selection
Some Notes on Adverse Selection John Morgan Haas School of Business and Department of Economics University of California, Berkeley Overview This set of lecture notes covers a general model of adverse selection
More informationBasics of Game Theory
Basics of Game Theory Giacomo Bacci and Luca Sanguinetti Department of Information Engineering University of Pisa, Pisa, Italy {giacomo.bacci,luca.sanguinetti}@iet.unipi.it April - May, 2010 G. Bacci and
More informationTheory of Auctions. Carlos Hurtado. Jun 23th, Department of Economics University of Illinois at Urbana-Champaign
Theory of Auctions Carlos Hurtado Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Jun 23th, 2015 C. Hurtado (UIUC - Economics) Game Theory On the Agenda 1 Formalizing
More informationCapacity Constraints as a Commitment Device in Dynamic Pipeline Rent Extraction
Capacity Constraints as a Commitment Device in Dynamic Pipeline Rent Extraction Lucia Vojtassak, John R. oyce, Jeffrey R. Church # Department of Economics University of Calgary Calgary A TN N4 Canada September
More informationTextbook Producer Theory: a Behavioral Version
Textbook Producer Theory: a Behavioral Version Xavier Gabaix NYU Stern, CEPR and NBER November 22, 2013 Preliminary and incomplete Abstract This note develops a behavioral version of textbook producer
More informationMixed oligopoly in a two-dimensional city y
Mixed oligopoly in a two-dimensional city y Takanori Ago z November 6, 2009 Abstract This paper analyzes a mixed oligopoly model with one public rm and two private rms in a two-dimensional square city.
More informationNonlinear dynamics in a duopoly with price competition and vertical differentiation
Nonlinear dynamics in a duopoly with price competition and vertical differentiation uciano Fanti, uca Gori and Mauro Sodini Department of Economics and Management, University of Pisa, Via Cosimo Ridolfi,
More informationControlling versus enabling Online appendix
Controlling versus enabling Online appendix Andrei Hagiu and Julian Wright September, 017 Section 1 shows the sense in which Proposition 1 and in Section 4 of the main paper hold in a much more general
More informationWars of Attrition with Budget Constraints
Wars of Attrition with Budget Constraints Gagan Ghosh Bingchao Huangfu Heng Liu October 19, 2017 (PRELIMINARY AND INCOMPLETE: COMMENTS WELCOME) Abstract We study wars of attrition between two bidders who
More informationExercises - SOLUTIONS UEC Advanced Microeconomics, Fall 2018 Instructor: Dusan Drabik, de Leeuwenborch 2105
Eercises - SOLUTIONS UEC-5806 Advanced Microeconomics, Fall 08 Instructor: Dusan Drabik, de Leeuwenborch 05. A consumer has a preference relation on R which can be represented by the utility function u()
More information