Optimal Advertising to Consumers with Differentiated Preferences

Transcription

1 Optimal Advertiing to Conumer with Differentiated Preference Daniel Z. Li Durham Univerity Bart Taub Glagow Univerity January 6, 2014 Abtract We tudy the optimal advertiing of product attribute to conumer with differentiated preference, in the cae of a monopoly eller. Conumer valuation depend on the match between the product attribute and their preference. Informative advertiing ha the univeral effect of rik-reduction for the conumer, and the idioyncratic effect of improving the match between the product and the target conumer. We firt provide a cloe-form olution on the relationhip between ignal preciion and the ditribution of conumer poterior valuation. We then prove that advertiing drive the clockwie rotation of the demand curve, and the demand curve atify the ingle-croing property, which implie that for more precie ignal, the optimal price are higher. And the profit maximizing information can be intermediate, which i in contrat to the reult in the literature. Furthermore, we alo how in numerical analyi that ocial welfare lo under monopoly tem from two ource: one i the monopoly price, and the other i information, a the monopoly eller generally reveal le information than ocially optimal level. And a tandard ubidy cheme by a regulator, that induce equal-to-marginal-cot price, will encourage the monopoly eller to reveal even le information. 1 Introduction Firm incur expenditure on advertiing in order to affect the demand for their product. Giving too little information via advertiing can lead rik avere conumer to hun the product; too much information can lead conumer to reject a product a unmatched with their preference. What i the optimum level of advertiing, and how i the price and profit affected? A product i normally characterized by both it vertical and horizontal attribute. Conumer hare the ame preference for vertical attribute uch a quality; revealing information about vertical attribute will hift the demand curve upward or downward, that i, it may increae or decreae the willingne to pay WTP for all the conumer. Daniel Z. Li, Department of Economic and Finance, Durham Univerity Buine School, Durham DH1 3LB, UK. daniel.li@durham.ac.uk, Tel: Bart Taub, Adam Smith Buine School, Univerity of Glagow, Bart.Taub@glagow.ac.uk. 1

2 On the other hand, conumer have differentiated preference for a product horizontal attribute, uch a deign or colour of a product. Advertiing the horizontal attribute may increae the WTP of ome conumer, while reduce thoe of other. A a reult, advertiing a product horizontal attribute will rotate the demand curve clockwie. The literature place advertiing into three categorie according to it function, a ummarized in Bagwell [2]. The firt i informative advertiing, which increae the awarene of conumer about the exitence of a product or it ubtitute, and therefore hift the demand outward or make it more elatic, and i therefore pro-competitive. The econd i peruaive advertiing, that reveal product attribute to conumer, increae each given conumer WTP and foter product differentiation. Peruaive advertiing reduce the elaticity of demand, that i, it teepen the demand curve, and reult in higher price and concentrated market. The third category i complementary advertiing, which hold that advertiing directly enter a conumer utility in a complementary way. For example, conumer may care about ocial pretige, and hi utility of conumption of a product may get higher if that product i well advertied. Focuing more on the direct impact of advertiing on demand, Johnon and Myatt [11] introduce a new taxonomy of hype and real information advertiing. Hype advertiing highlight the exitence of a product, and promote the vertical attribute of the product that are unambiguouly valuable. Thi type of advertiing hift the demand curve outward. Real information advertiing ignal a product horizontal attribute to conumer. A a reult, ome conumer will learn that the product i a poor match for them, but other conumer will realize that it i a good match and their WTP will increae. Conequently the demand curve rotate clockwie, which we will how may have ambiguou implication for a monopolit profit and the ocial welfare. There are a number of empirical tudie demontrating the rotation of demand under advertiing. Erdem et al [7] ue Nielen canner panel data of four categorie of conumer good to invetigate how advertiing affect the demand curve. For the three categorie of toothpate, toothbruhe and detergent, they find that advertiing increae the price elaticity of demand flattening the demand curve and lower the equilibrium price, which i conitent with the theoretical conjecture of Becker and Murphy [3]. However, they alo find an exception in the cae of Heinz ketchup, where the empirical reult how that advertiing decreae the price elaticity of demand making the demand teeper. They argue that thi i becaue it i horizontal attribute that are advertied in the cae of Heinz ketchup, while in the other three cae, the advertiing focue on vertical attribute. Rickard et al [14] invetigate the demand hift and rotation effect of advertiing in the American fruit and vegetable indutry, with a focu on conumer heterogeneity. Firt, they find that generic advertiing program induce both an outward hift and clockwie rotation of the demand curve, which indicate that the promotion campaign lead to an increae in the diperion of conumer valuation the rotation effect and increae the WTP among marginal conumer the hift effect. In addition, broad-baed advertiing alo hift the demand outward, yet it appear to have a counter-clockwie rotational effect on the demand. Thi i becaue broad-baed advertiing make conumer more aware of product ubtitute and thu i pro-competitive. 2

3 Anand and Shachar [1] tudy the informational role of the televiion advertiing in the US in matching conumer with product. They how that advertiing i informative and ha ignificant impact on conumer behavior. They alo how that expoure to advertiing may actually deter conumer purchae when they find a product i not a good match for them. On the other hand, advertiing ignificantly improve the match between conumer and product by guiding them to product that better fit their preference. 1 There have been everal theoretical tudie of the effect of information dicloure and advertiing on demand rotation Lewi and Sappington [12]; Johnon and Myatt [11]; Ganuza [8]; Ganuza and Penalva [9]. Though different in model et-up and in term of information tructure, 2 they commonly emphaize the ame intuition that more precie information will caue conumer poterior valuation to become more dipered. The intuition i that revealing a pecific product attribute may increae ome conumer WTP, yet may reduce that of other, and therefore conumer poterior valuation become more dipered. And advertiing product attribute to conumer with differentiated preference will induce the clockwie rotation of the demand curve. Demand rotation ha an ambiguou impact on profit. Firm face a trade-off between a niche and a ma marketing trategy. With a niche marketing trategy, a firm reveal precie information and create product differentiation, increaing the valuation of ome conumer. In thi cae, the firm will charge high price, while only a portion of the conumer are erved. With a ma marketing trategy, the firm reveal little product information and charge a low price, conequently erving more conumer. With thi baic trade-off, the common concluion of thee paper i that, when information i cotle, optimal dicloure policy i extreme, in the ene that the eller hould reveal either full or no information to conumer. Although preference differentiation eem pivotal for the reult of extreme dicloure policy, the theoretical literature ha o far not explicitly modeled preference, and ha not explicitly modelled ignal extraction by conumer. The prior literature adopt mainly a reduced-form approach by auming a monotonic relationhip between ignal preciion and the degree of diperion of poterior valuation, while remaining ilent on the underlying mechanim by which thi diperion come about. An exception i Ganuza [8], who explicitly model preference differentiation on a Hotelling circle, and invetigate optimal dicloure in an auction with cotly information, yet he doe not characterize the ditribution of conumer poterior valuation. We depart from the reduced-form approach and invetigate the optimal advertiing problem for a monopoly eller, where conumer valuation are endogenouly determined. Conumer preference are horizontally differentiated, and their valuation of the product depend on the matching between their preference and the product attribute. Therefore a particular product attribute may jut appeal to ome, but not all of the conumer. 1 Other reearch, include Zheng et al [16], that tudie demand rotation effect in the US non-alcoholic beverage market, find that doubling advertiing expenditure lead to clockwie rotation of the demand curve for milk and alo for coffee and tea, with lope change of 7% and 12% repectively. 2 Variou information concept are introduced, uch a rotation order in John and Myatt [11] and preciion order in Ganuza and Penalva [9]. 3

4 The conumer in our model are rik avere. They rationally calculate the poterior ditribution of the product attribute baed on their prior knowledge and the information revealed by the eller, and then optimize their purchae deciion. The eller pre-commit to an announced advertiing policy, and end public ignal in the form of additive Gauian noie, controlling ignal preciion by chooing the variance of the noie. Revealing information ha two effect. Firt, it ha a univeral effect of rik reduction for all the conumer. Second, it ha an idioyncratic effect by revealing the conditional expectation of the product attribute. It reaure conumer whoe type cloely matche the product attribute, but drive away conumer who realize that they are poorly matched. With thi framework, we firt derive a cloed-form olution on the relationhip between ignal preciion and the ditribution of conumer poterior valuation. Furthermore, we how that information dicloure induce the clockwie rotation of the demand curve. Specifically, correponding to different ignal preciion, the demand curve obey a ingle-croing property. Thi ingle-croing property alo implie that, for more precie ignal, the optimal price et by the monopoly eller i higher. In contrat to the extreme dicloure policy reult in the previou literature, we how that, depending on the ditribution of product attribute and conumer preference, it can be optimal for the eller to reveal intermediate level information. Furthermore, we numerically analyze ocial welfare with our model etting, and how that, under information dicloure, welfare lo under monopoly tem from two ource: the firt i the uual inefficiency temming from the monopoly price, which i higher than marginal cot of production, the ocially optimal level. The other i information: the monopoly eller generally ha intention to withhold information in the ene that he reveal weakly le precie information than ocially optimal level. Finally, we invetigate the tandard problem of monopoly regulation. In our model of complete information and no ditortion cot of public fund, we how that the conventional policy intrument of ubidy won t achieve the firt bet outcome of the economy. Our reult demontrate that, though ubidy drive the monopoly eller to et a price at the ocially optimal level, yet he will reveal weakly le precie information than ocially optimal level. The remaining part of thi paper are organized a below: Section 2 i model et-up; Section 3 tudie the benchmark cae of no information dicloure; Section 4 invetigate how advertiing induce the clock-wie rotation of the demand curve, and prove the ingle-croing property; Section 5 i about optimal pricing and advertiing trategy of the monopoly eller, with comparative tatic reult; Section 6 i an numerical invetigation of welfare loe under monopoly with advertiing; Section 7 invetigate the problem of monopoly dicloure under ubidy; and Section 8 i a hort concluion. 4

5 2 The model A profit-maximizing monopoly eller ell a product to a continuum of conumer, whoe preference are horizontally differentiated, indexed by the parameter θ. The product i characterized by it vertical value V and horizontal attribute. The vertical value V i commonly known, but the horizontal attribute i a random variable, whoe realization i known only by the eller. The population of conumer i normalized to one, and each conumer ha a unit demand, ay buying either one or zero unit of the product. A conumer valuation of the product depend on the proximity of the product attribute and hi own preference parameter θ, which conform to a ditribution of F θ. The vnm utility function of a type θ conumer i u, p; θ = V τ θ 2 p 1 where p i the price and τ a coefficient meauring diutility of the degree of mimatching. 3 worth noting that conumer are rik avere in, the realization of the horizontal attribute. Let conumer reervation utility be 0, and a type θ conumer will buy one unit of product if and only if the expected utility Eu, p; θ 0. The product attribute ha a Gauian ditribution with a known mean and variance, pecifically N0, σ 2, and we denote the cumulative ditribution function of a Φ. It i Both ditribution of and θ are commonly known, but the realization of the product attribute and the conumer preference θ are privately oberved jut by the eller and the correponding conumer. Beide etting the price, the eller alo decide how much product information to reveal to the conumer, by ending a public ignal of m in the following form m = + ν 2 where ν i an uncorrelated Gauian noie, pecifically ν N0, σν. 2 Therefore, m N 0, σm 2 with σ 2 m = σ 2 + σ 2 ν. For given ditribution of the product attribute, the dicloure policy of 2 i fully characterized by σ 2 ν. 4 The maller the σ 2 ν, the more precie the ignal i. For convenience, we introduce an 3 Our formulation of preference i imilar to that of Sun [15] in that there i a vertical attribute and a horizontal attribute, and the proximity of the horizontal attribute to the product characteritic partially determine the conumer payoff. However, we ue quadratic rather than abolute-value payoff, jut a in Ganuza [8]. There are ome imilaritie with the model of Drugov and Troy-Martinez [6] a well, in that the eller end a ignal that i a noiy verion of the true product attribute. Where they deviate from our approach i that the eller can control not only the noie in the ignal, but alo the mean they call thi bia. They aume that buyer know the variance of the noie a we do alo but that they cannot oberve the bia directly. However, direct ignal extraction i ignored in the model. Even though we don t have bia in our model, we do have direct ignal extraction about the realized attribute conditional on the ignal. 4 The previou literature poit a matching technology between advertiement and conumer. For example, in Bergemann and Bonatti [4], building on the earlier model of Butter [5], a et of meage i matched with the et of conumer, and there i a probability of a match between a particular conumer and a particular meage. If there i a match, the conumer i aumed to become aware of the product. By contrat, we aume that there i a ingle common meage given to all conumer. Becaue it i the um of Gauian random variable, it i Gauian: becaue 5

6 equivalent meaurement of ignal preciion, which i defined a σ2 σ 2 + σ 2 ν 3 It i evident that = 0 correpond to an infinitely noiy ignal, and = 1 correpond to a perfectly informative ignal. We alo aume that it i cotle to end the ignal, which allow u to iolate purely informational effect from cot-induced effect. 5 Chooing the ignal variance σ 2 ν i equivalent to chooing the ignal preciion, and we will denote the ditribution of the ignal conditional on by a cumulative ditribution function of Φ. The timing of the game i a follow: 1. The eller announce a dicloure policy in the form of 2, and et the price p for the product; 2. Nature elect the realization of and θ, which are then privately oberved by the eller and the correponding conumer repectively; 3. Oberving, the eller end a public ignal of m according to 2; 4. Receiving m and knowing the price p, conumer make their purchae deciion. Upon receiving a public ignal of m, the conumer will update their belief and form poterior valuation of the product. Let Bp, m denote the et of conumer type with poitive demand, then from 1 it i obviou that B p, m = { θ : V τe [ θ 2 m ] p } Baed on the propertie of Gauian ditribution, we have the following reult on Bp, m. Lemma 1 Conditional on a ignal m and price p, the et of conumer type with poitive demand i B p, m = {θ : θ m p, }. 4 V p where p, = τ 1 σ. 2 Proof. To olve the problem it i neceary to calculate the conditional expectation. The joint ditribution of m i alo Gauian: m N µ µ m the product characteritic and the noie are both Gauian it i optimal for conumer to extract the ignal of the product characteritic via a imple linear projection. 5 Other literature, uch a Bergemann and Bonatti [4], note the excluivity of the attention of conumer on a ingle meage at a time, leading to cotly expenditure to acquire that attention., σ 2 σ 2 σ 2 σ 2 m 6

7 a any linear combination of, m i Gauian. Furthermore, the conditional ditribution of m i Gauian a well Greene [10], page 1013 Theorem B.7, with conditional mean σ2 E m = µ + σ 2 + σv 2 m µ m = m and conditional variance Therefore we can compute var m = σ 2 σ2 σ 2 σ 2 + σ 2 v = 1 σ 2. E [ 2 m ] = var m + E 2 m = σ m 2 Conditional on m, the et of conumer type with poitive demand i, { [ θ : V τe θ 2 m ] p } = { θ : V τ E [ 2 m ] 2θE [ m] + θ 2 p } = { θ : V τ σ m 2 2θm + θ 2 p } yielding 4. Equivalently, the type et of 4 can be re-written a { θ : θ [ ]} θ p, ; m, θ p, ; m where the two threhold are θ p, ; m = m p, θ p, ; m = m + p,, 5 It i intereting to oberve that m = E m, the conditional mean of product attribute, and 1 σ 2 = var m, the conditional variance of product attribute. So the et of Bp, m, a an interval, i centred around m, the conditional mean of the product attribute, and expand to a range of, which depend on the conditional variance of the product attribute. Advertiing policy i characterized by ignal preciion, which affect the et of Bp, m through two way. Firt, increaing reduce var m, the conditional variance of the product attribute and thu the rik for all the conumer, widening the threhold of p,. Second, increaing alo get the conditional mean of the product attribute, m, to be more enitive to ignal. Holding the ignal m fixed, thi may move the center of ma of demander outward toward thoe with eclectic tate, who are le numerou. Due to thee two competing effect, increaing can either increae or reduce demand. The threhold factor p, hrink in σ, 2 the unconditional variance of the product attribute, which alo repreent the ex ante rik facing the conumer. A the conumer are rik avere, increaing σ 2 will narrow the et of conumer with poitive demand. And apparently, increaing price p unambiguouly hrink p, and thu the demand. In thi paper, we aume conumer preference parameter θ i exponentially ditributed along the poitive real line: fθ = 1 e θ/ and F θ = 1 e θ/ 0 < θ <. 6 7

8 A imilar aumption of exponential ditribution i alo made in Bergemann and Bonatti [4], and thi etting allow u to tractably calculate equilibrium quantitie. 6 In our ubequent analyi it will be ueful to ditinguih between the interim expected demand, which i conditional on a pecific ignal m, and the ex ante expected demand. Conditional on a ignal m, the interim expected demand for the product i θp,;m 0 m < / d p, ; m = df θ = F θ / m / θp,;m F θ 7 F θ m / where = p,. And the ex ante expected demand i jut D p, = d p, ; m dφm = / F θ dφm / F θ dφm 8 Exploiting the aumption of exponential θ, we are able to derive a cloe-form repreentation of the ex ante demand function: Lemma 2 Given the exponential aumption of 6 and that 0, 1], the ex ante expected demand function i D p, = [ Φ Φ ] + exp σ 2 [ 2 2 exp σ Φ σ 2 Proof. See the Appendix. + [ 1 Φ σ 2 + ] ] 9 Under our etting of Gauian ignal and exponential preference, Lemma 2 provide a cloeform olution that explicitly reveal the relationhip between ignal preciion,, and the ditribution of conumer poterior valuation, repreented by the ex ante demand function Dp,. The demand function of Dp, i determined by price p and ignal preciion. The eller problem i to maximize expected profit, max πp, = p [0,V ], [0,1] p dp, ; m dφm = pdp, 10 Holding ignal preciion fixed, the olution of optimal price p to problem 10 exit baed on the maximum value theorem. And it i immediate that the optimal price mut be interior, that i, p 0, V, becaue if p = 0 profit i zero, and when p = V expected demand i zero. And an intereting quetion i whether the optimal ignal precion,, i interior or not. 6 The exponential ditribution ha a ingle parameter,, with mean and variance 2 : we cannot therefore carry out numerical experiment in which there i a mean preerving pread of preference, becaue the variance increae with the mean. However, we can effectively do thi indirectly by altering the variance of the firm product characteritic, σ 2. 8

9 3 Benchmark: No dicloure and full dicloure To develop intuition, we tart with the benchmark cae of no information dicloure = 0, and then compare it numerically with the other extreme cae of full information dicloure, where the eller truthfully reveal the product attribute and = 1. In the benchmark cae of = 0, the et of conumer type with poitive demand i B p, 0 = {θ : θ p, 0} from 4. And the ex ante expected demand i imply D p, 0 = θp,0 θp,0 df θ = F θp, 0 11 For 0, 1], we already derive the ex ante demand D p, in 9, and it intereting to oberve that D p, 0 = lim 0 D p, The econd obervation i that, when σ 2 V τ, then no trade can happen at any p > 0 in the benchmark cae of = 0, a p, 0 i not well defined in thi cae. When there no information dicloure, σ 2 meaure the rik facing the conumer, if it outweigh the net value of the product V τ, then no conumer will buy the product at any poitive price. Lemma 3 In the cae of no information dicloure, if then no trade will happen at any price p > 0. σ 2 V τ Thi reult imply tate that, when there i no information dicloure, if the variation of the product attribute i high enough, no conumer would buy the product at any poitive price, due to rik averion. When σ 2 < V, olve profit maximization problem and we get the following firt τ order condition D p, 0 D p, 0 + p = 0 p From the expreion of D p, 0 in 11, it i eay to get the optimal price a p 0 = 2τ p, 0 1 e p,0/ e p,0/ = 2τ p, 0 F θ f θ Let π0, τ, V denote the correponding maximum profit function, where the ubcript 0 repreent the cae of = 0, and we have the following comparative tatic reult: Lemma 4 π 0 i trictly decreaing in, π 0 < 0 9

10 Figure 1: Demand Rotation and Profit = Β 1, V 10, Τ 1, Σ 2 1 Α 0 b, Α 0.5 d, Α 1 r Demand Β 1, V 10, Τ 1, Σ 2 1 Α 0 b, Α 0.5 d, Α 1 r Profit [ Proof. By the envelope theorem, π 0 = p D 0 = p 0 p,0 ] 0 e p 2 0,0/ < 0. The intuition i traightforward: meaure both the mean and the variance of the exponential ditribution of conumer preference. When increae, the ma of conumer preference move farther away from that of product attribute, implying that le conumer are intereted in the product, and therefore demand and maximum profit will decreae. We contrat the behavior of the no-dicloure and full-dicloure cae in the following numerical example. In the firt example, the left panel of Figure 1 illutrate the behavior of the ex ante demand Dp,, with an analytical expreion of 9. The two extreme cae of = 0 and 1, together with the intermediate cae of = 0.5, correpond to the blue, red and dahed curve repectively. It i worth attention that the axe are revered relative to the uual portrayal of demand curve: price i on the horizontal axi, quantity on the vertical axi. A hown in thi panel, with increaingly more precie ignal the demand curve rotate relative to the no-dicloure demand curve o a to become le elatic at the pivot point. The right panel of Figure 1 how profit a a function of the price, given the full-dicloure and non-dicloure ignal. In thi example, withholding information = 0 generate higher profit than that of full information dicloure = 1. And the optimal price under full dicloure i higher than that under no information dicloure. In the econd example, illutrated in Figure 2, the demand parameter ha been increaed from 1 in Figure 1 to 1.5. Thi increae both the mean and the diperion of conumer preference, that i, it drive the ma of the ditribution of conumer preference farther away from 0, the expected product attribute. Going from no dicloure to full dicloure again rotate the demand curve, making it le elatic at the pivot point. But thi time the maximum profit i attained by full dicloure. The profit-maximizing price for full dicloure i again higher than that for no dicloure. 10

11 Figure 2: Demand Rotation and Profit = Β 1.5, V 10, Τ 1, Σ 2 1 Α 0 b, Α 0.5 d, Α 1 r Demand 0.4 Profit Β 1.5, V 10, Τ 1, Σ 2 1 Α 0 b, Α 0.5 d, Α 1 r Information dicloure and demand rotation The example in Figure 1 and Figure 2 ugget that revealing more precie information rotate the demand curve clockwie. In other word, correponding to different ignal preciion, the demand curve atify ome ingle-croing property. We alo oberve that, in both example, the optimal price under full dicloure i higher than that under no information dicloure. Thi ugget that there i a monotonic relationhip between ignal preciion and the optimal price, which we will how i an implication of the ingle-croing property. In thi ection, we invetigate the general relationhip between ignal preciion and demand rotation, and provide a formal proof of the ingle croing property, which tate that, for any pair of ignal with different ignal preciion, the correponding demand curve cro only once. Unlike Johnon and Myatt [11] who introduce the ingle croing property a an aumption, we prove it in our model with conumer endogenou valuation how information dicloure drive the rotation of demand curve. Before the proof of the ingle croing property, we firt provide ome reult on the comparative tatic of the demand curve of Dp,. The firt reult i that, for given, Dp, i both trictly decreaing and concave in price. Lemma 5 The ex ante expected demand, D p,, i both trictly decreaing and trictly concave in p, that i Proof. See the Appendix. D p, p < 0 Specifically, we have the following expreion [ D p = 1 2τ exp σ Φ σ2 2 D p, p 2 < 0 + exp σ ] Φ σ2 where = p,. Furthermore, the trict concavity of Dp, in p alo implie that, for given, the expected profit πp, = pdp, i alo trictly concave in price. Thu, for any given ignal 11 12

12 preciion, there exit a unique interior price that maximum the expected profit. The behavior of demand with repect to ignal preciion i le traightforward. Ideally, we would like to etablih that demand behave monotonically in ome ene a preciion increae. Thi i not true in general, but we can at leat tate ome formula and decompoition that can be ued in the ubequent analyi. We begin with the following reult. Lemma 6 [ ] D = σ2 2 2 D p, + 1 2Φ τσ 2 D p 13 Proof. See the Appendix. The impact of a change in ignal preciion on Dp, i twofold, a hown in 8: firt, a change of change the ditribution of ignal, Φm, a the variance of ignal σ 2 m = σ 2 /; econd, changing alo change the interim demand d p, ; m by changing the value of the threhold, θ and θ, a hown in 7. And the net effect i ambiguou. However, increaing ignal preciion will unambiguouly reduce the rik facing all the conumer, and will increae the highet poible value of conumer WTP. Let p denote the highet price chargeable at, and it i evident that which i trictly increaing in. p = V τ 1 σ 2 14 We provide two numerical example illutrating the repone of demand to change in the ignal preciion in Figure 3. The left panel plot the repone of demand to ignal preciion for the cae of = 1, and the right panel plot the repone for the cae of = 3. In both cae, we oberve that for given, the partial derivative of D croe the zero axi from negative to poitive once, reflecting the ingle-croing property. Figure 3: Demand repone to preciion Β 1, V 10, Τ 1.0, Σ 2 1 Α 0 b, Α 0.5 d, Α 1 g Β 3, V 10, Τ 1.0, Σ 2 1 Α 0 b, Α 0.5 d, Α 1 g D Α 0.02 D Α Next we would like to etablih the formal reult that the demand curve rotate monotonically when increae, which i in turn a conequence of the ingle croing property. Thi property i 12

13 defined a rotation order by Johnon and Myatt [11], and their definition of rotation order i a follow: Definition 7 Rotation Order ingle croing property: For a local change of, the demand curve rotate around ome price ˆp, that i p ˆp D p, 0 Here D ˆp,, ˆp i the rotating point, which i -pecific. The definition imply tate that for a local change of ignal preciion, the demand curve will rotate clockwie around the -pecific rotating point. And it implie that, evaluating at price p, the demand become le elatic when increae. Demand rotation i implied by the ingle croing property of the demand curve. A caveat i that if we require that p mut be poitive, then the rotation point ˆp, D ˆp, may not exit. And a more general definition of the ingle croing property i: Definition 8 The ex ante demand function D p, ha the ingle croing property, if whenever > and p > p, then D p, D p, > 0 D p, D p, > 0 Thi definition of ingle-croing property i more general than that of Johnon and Myatt [11]. According to thi definition, if Dp, atifie the ingle-croing property, then for any >, then Dp, Dp, will cro the zero-line from negative to poitive at mot once. So if a poitive rotating point doe not exit, we will alway have Dp, > Dp,, and if it exit, then we revert to the definition of Johnon and Myatt. In addition, the change of i not retricted to local change in our definition. Propoition 9 The ex ante expected demand function D p, atifie the ingle-croing property. Proof. See the Appendix. Here we etablih one of the major reult of thi paper. In a model with explicit ignal and conumer endogenou valuation, we how that advertiing product information to conumer with differentiated preference will drive the clockwie rotation of the demand curve, a implied by the ingle-croing property. Thi property alo implie that more precie ignal will reult in more dipered ditribution of conumer poterior valuation. In the previou literature, thi property i introduced a an aumption, while here we prove it in a model with conumer endogenou valuation. A direct implication of the above Propoition i that the profit function, πp, = pdp,, alo atifie the ingle-croing property. Corollary 10 The ex ante expected profit function π p, atifie the ingle-croing property. 13

14 Proof. A π p, = pdp,, the proof i elf-evident. The ingle croing property ha two implication. The firt i that when increaed dicloure induce rotation of the demand curve, the elaticity of the demand curve necearily decreae. With the concavity reult of Lemma 5, thi decreae i global. Thi in turn implie that it i optimal to increae the price relative to the optimal price for the unrotated curve. Increaed dicloure thu lead to an increae in price. Becaue increae in price reult in a reduction of quantity demanded in equilibrium, it i correct to label etting with high price a niche market. 5 Optimal pricing and information dicloure In thi ection we combine our previou reult to imultaneouly olve for optimal price and ignal preciion for the eller, and analyze how they repond to change in the parameter. Optimal pricing By Lemma 5, for given, demand D p, i trictly concave in p; it i immediate that profit i concave a well. Therefore, for given, there exit a unique optimal price p that maximize expected profit p = D D/ p = 2τ / F θ dφm / F θ dφm / f θ dφm + / f θ dφm 15 and p i interior. Furthermore, the ingle-croing property of πp, implie a monotonic relationhip between the optimal price p and the ignal preciion. Propoition 11 The optimal price p i increaing in, that i, for > p > p Proof. It i eay to demontrate that π p, i quai-upermodular in p. By Corollary 10 it and atifie the ingle-croing property in p,. The reult follow from applying Theorem 4 in Milgrom and Shannon [13]. The reult alo reveal the trade-off between a ma and a niche marketing trategy facing the monopoly eller. For ma marketing trategy, he reveal little product information to the conumer, he hould charge a low price while have a large demand. While for niche marketing trategy, he will reveal more precie information and charge a high price, and only thoe conumer who find it i a good match for them will be erved. Between the niche and ma marketing trategy, which one i better depend on the optimal degree of information dicloure, which i in turn determined by the ditribution of the product attribute and conumer preference. Optimal dicloure We now explore the eller problem of optimal information dicloure. The central quetion i whether optimal ignal preciion i extreme, that i, = 0 or 1, or intermediate, with 0, 1. 14

15 The numerical example in Figure 4 how that can be extreme. The two panel in Figure 4 plot the profit contour of price p and ignal preciion. The panel on the left how that when = 1, not revealing information = 0 generate the maximum profit. The panel on the right how that when i increaed = 3, full information dicloure = 1 i optimal. Figure 4: Optimal ignal preciion be extreme Profit Contour Β 1,Τ 1,V 10,Σ Profit Contour Β 3,Τ 1,V 10,Σ Α Α p p One common reult in the previou literature i that the optimal dicloure policy i extreme in the ene that, the eller will either reveal full or no information to the conumer. For example, in Johnon and Myatt [11], thi reult i baed on the aumption that with increaingly more precie ignal, the rotating point of D ˆp,, ˆp move to the right, that i, ˆp get lower when increae. Thi aumption implie that, the demand curve correponding to intermediate level preciion 0, 1 are necearily enveloped by the two of extreme cae = 0or 1, and therefore optimal dicloure mut be extreme. However, thi aumption i not univerally true in our model with endogenou valuation. For an intermediate value of of 1.45, the example in Figure 5 how that the optimal ignal preciion can be intermediate. Furthermore, from the ingle-croing property of the demand curve Dp,, we can provide a ufficient condition under which full information dicloure i optimal for the monopolit. Lemma 12 If D0, < D0, 1 for any [0, 1, then full information dicloure i optimal for the monopolit, that i, = 1. Proof. A the demand Dp, atifie the ingle-croing property, we then have Dp, < Dp, 1 for any price p 0. And therefore, full information dicloure, = 1, i optimal. One direct implication of the Lemma i that, for < 1, it i neceary that at price p = 0, there exit ome < 1 uch that D0, D0, 1. The numerical example how that when or σ 2 get bigger, that i, when the ma of conumer preference move farther away from the 15

16 Figure 5: can be interior Profit Contour Β 1.45,Τ 1.04,V 10,Σ Α p ditribution of product attribute, full dicloure become optimal for the monopolit. Roughly peaking, when the ditribution of conumer preference and the product attribute are far enough from each other, it i better for the eller to reveal full information to the conumer. The intuition i that, if no information i provided in thi cae, mot conumer will expect that the product attribute i not a good match for them and a no information i provided, the rik of mi-matching i high. A a reult, both the demand and the WTP are low. If the eller provide precie product information in thi cae, then it reduce the rik facing the conumer, and he i able to charge high price for a maybe mall portion of conumer who find it a good match for them. Comparative tatic In thi ubection we analyze how optimal dicloure and optimal pricing repond to change in the parameter that determine the ditribution of conumer preference and product attribute. An increae in increae the mean and diperion of conumer preference; the increae in the mean mean that the average conumer get further away from the mean of the product attribute, and increaed information to overcome the rikine of a purchae. An increae in σ 2 increae the diperion of the product characteritic, thu making the purchae rikier for uninformed conumer. For informed conumer, the rik i eliminated but the firm face pricing rik. Figure 6 how the optimal repone of dicloure and price to the change of the parameter. In the left panel, when both σ 2 and are mall, meaning that the product attribute and conumer preference are highly concentrated and near to each other, it i optimal to withhold the information = 0. However, when both σ 2 and increae, a threhold i reached, beyond which full dicloure become optimal. There i an intermediate range around the threhold in which partial dicloure i optimal. 16

17 The optimal price alo repond to the σ, 2 threhold. Above the threhold where there i full dicloure, price hrink a σ 2 increae. Thi reflect the firm need to bring more cutomer in a the volatility of the product attribute increae, with fewer cutomer located cloe to the attribute. Below the threhold, price i dicretely lower relative to the continuation of the full-dicloure pricing manifold. Thi reflect the rik premium that firm offer conumer via lower pricing, given that there i le information dicloure. For mall value of, conumer are highly clutered near the product attribute; many conumer will buy even with little information dicloure becaue the rik i low. Figure 6: Optimal dicloure and pricing The left panel plot optimal dicloure and the right hand panel plot the aociated optimal price for a range of and σ 2 parameter. Holding σ 2 fixed at low value, a increae, optimal dicloure increae. We can characterize the behavior of dicloure and price in thee figure in relation to our earlier ma market and niche market notion. To reiterate, ma market are characterized by low price and le than full dicloure, with a large number of conumer purchaing the good, whilt niche market are characterized by greater dicloure and higher price, reulting in fewer conumer purchaing the good. Thi correlation i apparent in the figure in two dimenion. In the firt dimenion, a we increae product attribute volatility σ, 2 there i a rapid jump in dicloure from no dicloure to full dicloure at a value of around At thi threhold, price take a dicrete jump upward a well. In the econd dimenion, for low value of σ 2 where there i partial dicloure, a the conumer preference parameter increae, the degree of dicloure increae, and o doe price. Thi effect i far le pronounced than it i in the harp tranition that occur in the σ 2 dimenion, but the poitive correlation of dicloure and price i till apparent. 17

18 6 Welfare We next invetigate the welfare implication of our model where a monopoly eller make the joint deciion of pricing and information dicloure. We define ocial welfare athe um of the eller profit and conumer urplu. 7 We have already pecified the profit of the eller. We calculate expected conumer welfare a the integral under the demand curve given the optimal price and information preciion, CSp, = p p Dx, dx where p = V τ 1 σ 2 i the maximum price chargeable for. And the ocial welfare i thu the um of the expected conumer urplu and the eller profit, which i equal to { W p, = V τ[1 σ 2 + θ m 2 ] } df θdφm Socially optimal price Bp,;m A a benchmark for our analyi of the monopoly equilibrium we etablih the ocially optimal price and dicloure. A there are no cot of production, the ocially optimal price hould be zero, and thi i the cae. Propoition 13 The ocially optimal price, p = 0. Proof. To pare notation, define the inner integral of W p, a w p,, that i, w p, = Applying Leibniz formula, wp,,m p = 1 = θp, θp, 1 [ V τ 1 σ 2 τ θ m 2] df θ [V τ 1 σ 2 τ θ m 2 ] [ exp exp [ V τ 1 σ 2 τ θ m 2] exp ] θ + exp θ p θ p 0 θ θ p θ θ p and the inequality i trict for p > 0. Therefore, W p, p = [ ] p w p,, m φmdm = wp,,m p φmdm 0 And therefore the ocially optimal price i zero. 7 Note that conumer urplu i not quite the ame thing a the direct payoff to conumer, which are not commenurable with profit. 18

19 Figure 7: Social Welfare: W p, Α Socially optimal dicloure When price i et at the ocially optimal level of zero, the ocial welfare i jut equal to the area below the demand curve, D0,. Full dicloure doe not necearily maximize ocial welfare, becaue dicloure rotate the demand curve, and thi can either increae or decreae total urplu, depending on the parameter. Denote the ocially optimal dicloure by. When p = p = 0, the marginal ocial value of dicloure i W p, = p 0 D x, dx We etablih from numerical example that thi expreion cannot be uniformly igned, and that can be zero or interior depending on the value of the parameter ee Figure 7. The monopolit of coure raie price above zero, and thi alone would hrink welfare, holding information dicloure fixed. In addition, the monopolit weakly hrink, thu reducing dicloure relative to the ocial optimum. We preent ome numerical example in Table 1 and 2. Our objective i to compare the ocially optimal welfare with that under monopoly in different model, which are defined by a et of parameter σ, 2, V, τ. The optimal ignal preciion and price choen by the monopoly eller are p and. Correpondingly, the optimal profit and ocial welfare are denoted a π = πp, and W = W p, repectively. A defined before, p and are the ocially optimal price and ignal preciion, we denote the ocially optimal welfare a W = W p,. In Table 1, it i alway ocially optimal to fully dicloe the information to conumer, but the monopolit chooe le-than-full preciion of the ignal in the lower range of product quality volatility. A the volatility of the product attribute increae, preciion increae and price decreae. In thi range of volatility, purchaing the product i riky. However, it i not optimal for the firm to eliminate the rik entirely by ending a fully informative ignal; rather, it partially increae the preciion and alo lower the price to compenate for the rik, drawing in more conumer. A the volatility of the product attribute increae, there i ome non-monotonicity in the 19

20 σ 2 p π W π W % W W W W % Table 1: Socially optimal and monopoly dicloure and welfare = 1.5, V = 10, τ = 1 repone of price. At the bottom of the table there i full dicloure, o the decreae in price reduce profit directly, but conumer welfare alo hrink due to the impact of the increaed rik from the higher product attribute volatility. 8 Table 2 provide a econd et of reult for a different et of parameter, varying. Increaing roughly mean moving the ma of conumer preference farther away from the ditribution of the product attribute. An intereting obervation i that withholding information could be ocially optimal, a hown in the firt three row of the table. Thi eemingly counter-intuitive reult tem from the fact that information dicloure in our model derive the rotation, rather than hifting, of the demand curve. When the ditribution of the product attribute overlap maively with that of conumer preference, ay i mall, it better to withhold the information, a revealing it will reduce the valuation of a ignificantly large number of the conumer. A in the previou ection, we can derive a ufficient condition here under which full information dicloure i both ocially optimal and profit-maximizing for the monopolit, baed on the inglecroing property of the demand curve. And the intuition i the ame a before. Lemma 14 If D0, < D0, 1 for any [0, 1, then full information dicloure i both ocially optimal and profit maximizing for the monopolit, that i, = = 1. 8 Conumer urplu ha a very nonlinear repone to parameter change. For low value of that i, conumer tightly clutered near zero, with low information preciion an increae of product quality volatility σ 2 lead to decreaed conumer urplu. Thi i becaue with low ignal preciion, conumer face high rik from purchaing the product. For high value of, where conumer are dipered and far from zero, the firm end a high-preciion ignal, but alo raie price. The net effect can be to increae conumer urplu. A caveat i that thee welfare comparion are to ome extent comparing apple and orange: a we increae, we are not examining the impact of a price change on a fixed demand curve; rather, the demand curve itelf i hifted. Moreover, a increae beyond the threhold where full dicloure occur, the rotation of the demand curve induced by the dicloure alo hift the demand curve, and thi alone alter the potential conumer urplu that can be generated. Thu, conumer urplu change have two caue: the hift of the potential demand curve determined by the parameter, pecifically and σ, 2 and the rotation of the demand curve determined by the optimal for the monopolit dicloure, which in turn arie from profit maximization. However, the level of welfare portrayed in the plot are not really comparable with each other, becaue it i the demand curve itelf that hift. 20

21 p π W π W % W W W W % Table 2: Socially optimal and monopoly dicloure and welfare V = 10, τ = 1, σ 2 = 1 Proof. The ocially welfare i meaured by the area below the demand curve, that i, W p, = p 0 Dx, dx. From the ingle-croing property, we know that Dp, < Dp, 1 for any price p 0, and therefore full information dicloure i ocially optimal. 7 Optimal Dicloure under Regulation In thi ection, we examine the tandard problem of monopoly regulation. We aume that there i no ocial cot of public fund, ay there no ditortion cot of taxation. And the regulator objective i to maximize the ocial welfare, which i equal to the um of conumer urplu and firm profit. With thi etting, the ocially optimal price i a price equal to the marginal cot of production, which i zero. We aume the regulator purue that target only though taxe or ubidie, yet can t directly control the degree of information dicloure by the monopoly eller,. A in the tandard cae,a regulator provide ubidy to the monopolit that induce it to produce at the price-equal-tomarginal-cot point. In that cae, profit i p + hdp,, where h i the per-unit ubidy. For given, if there i a ubidy the monopoly firm firt order condition i D p, + p + hd p p, = 0 where D p p, i the partial derivative of the demand with repect to p. Becaue marginal cot i zero, the ubidized price mut be zero. We then have the optimal ubidy level, denoted a h : h D 0, = D p 0, 16 Compared with the optimal monopoly price p, we have the following reult: Lemma 15 For given ignal preciion, the optimal ubidy level h i higher than the optimal monopoly price p : h p 21

22 W W W W W W % Table 3: Socially optimal and monopoly dicloure under ubidy V = 10, τ = 1, σ 2 = 1 Proof. Firt, a p > 0, we have D0, > Dp, > 0. Second, from Lemma 5, we know 2 Dp, p 2 < 0, and therefore 0 > D p 0, > D p p,. Thi implie that the monopolit achieve higher profit level under the optimal ubidy cheme, a both h > p and D0, > Dp,. The next quetion i whether the ocially optimal dicloure i implementable by the ubidy cheme or not. We define the profit of the monopolit under the ubidy cheme of h a π = h D0, = D2 0, D p 0, where the ubcript repreent ubidy. The marginal contribution of increaing ignal preciion to π, conditional on the ubidy cheme h, i π = D0, D 2 p0, [2D 0, D p 0, D0, D p 0, ] where D p i the econd order cro derivative. For the regulator, when price i et at the ocially optimal level, p = 0, the marginal impact of ignal preciion on ocial welfare i W 0, = p 0 D x, dx which reveal a different incentive from that of the monopoly eller. Table 3 provide a et of reult on how the monopoly eller repond to the ubidy cheme of h, for a et of imilar parameter a Table 2. In thi table, i the optimal choice of ignal preciion by the monopolit under ubidy, and W i the correponding ocial welfare. Other notation are the ame a before. The firt obervation i that the ubidy cheme, h, can not alway induce the monopoly eller to reveal information at the ocially optimal level. Under the ubidy cheme, the monopoly eller ha incentive to reveal weakly le information than the ocially optimal level

23 The econd and omehow more triking obervation i that, compared with the cae of no regulation, a ubidy cheme of h will induce the monopoly eller to withhold even more information, that i, An explanation for thi reult could be a follow: Lemma 15 implie that, for given ignal preciion, the unit ubidy level of h i higher than that of optimal price p, and therefore, the monopoly eller ha tronger incentive to maintain a large demand for the product, which could be achieved by withholding information lower and puruing a ma marketing trategy. 8 Concluion We have developed a theoretical framework that tudie optimal advertiing of product attribute to conumer with differentiated preference, with conumer valuation determined by the match between the product attribute and their preference. A monopoly eller can decide how much information to reveal to the conumer, by chooing ignal preciion. Informative advertiing ha two effect: the univeral effect of rik reduction for the conumer, and the idioyncratic effect of improving the match between the product and the target conumer. We provided a cloed-form olution that how the relationhip between the preciion of advertiing ignal and the ditribution of conumer poterior valuation. We alo howed that advertiing product attribute caue the demand curve to rotate clockwie, and proved that, correponding to ignal of different preciion level, the demand curve aociated with each preciion level atify a ingle-croing property. A comparative-tatic implication of the ingle-croing property i that, for more precie ignal, the optimal price choen by the eller hould be higher. We alo we howed that profit-maximizing preciion can be intermediate, in contrat to the tandard reult of extreme dicloure in the literature. We analyzed ocial welfare numerically, and howed that the welfare lo under monopoly tem from two ource. One i the tandard lo from monopoly pricing that i higher than the ocially optimal level, and the other i information, in that the monopoly eller generally reveal weakly le information than i ocially optimal. Finally, we invetigate the tandard problem of monopoly regulation, and how that a ubidy cheme, which induce the monopoly eller to et a price equal to marginal cot of production, will encourage her to withhold even more information than in cae of no regulation. Reference [1] Bharat Anand and Ron Shachar. Avertiing, the matchmaker. RAND Journal of Economic, 422: , [2] Kyle Bagwell. The Economic Analyi of Advertiing, volume 3, page North- Holland: Amterdam,