 In the title we have this term optional intermediaries and what we mean by that is simply a platform that users can choose to use or not to and particular to make purchases. So the examples would be that if you're going to say you know you're interested in buying a water filter you can buy it via amazon or you can go directly to the seller's website and buy it from the seller or you want to order a meal from a particular restaurant you can order it using an app like DoorDash or you can call the restaurant or order directly from the restaurant's website. Similarly hotel rooms you know lots of examples like this and the pricing restraints that we're interested in are the so-called price parity clauses or as it's called in some of the literature price coherence. So we're thinking about this distinction between the case where the platform imposes a price coherence restraint versus the case where the platform allows the sellers to have price flexibility and we're asking some pretty basic questions like what effect did these restraints have on consumer surplus and on total surplus and when do platforms have an incentive to impose price coherence. So for some context these this this issue the issue of the effects of price parity clauses and so forth has a long-standing place in economic discussion and in policy discussions. There's it's related to issues of credit card payment card regulation another big issue that I'm sure people are familiar with has to do with most favored nation clauses and there there's often a story about competition and the effects that these might have on competition that's not going to be part of what what we look at. Another one that's gotten another area that's got gotten a lot of attention recently has to do with the rules that app stores set for software developers. So you know I'm not going to go into this too much because the goal in this paper is to remain quite abstract and to try to uncover a set of crucial driving forces in a pretty basic model of this situation and these are forces that at least to the best of our knowledge are interesting and understandable but also have not really received clear treatment or as much attention as we think would be useful. Now to preview a little bit the model is very directly inspired by the model that appears in a paper by Julian Edelman by Ben Edelman sorry Ben Edelman and Julian Wright in the QGE 2015 which in a sense our motivation for getting involved with this with this topic was because we found that paper I mean I saw Julian present that paper so Ben present that paper at Julian's conference a long time ago in Singapore and always found it to be really interesting and so it's particularly nice to have Julian as the discussant for for this and one of the main findings I think this is a fair characterization a main finding of that paper is that price coherence tends to be harmful because it leads to excessive intermediation it leads to as the authors would would probably put it too many transactions going through the platform compared to what would be efficient and in a sense as somebody who's flown all around the world buying economy class tickets and often upgrading to business class I found this story to be personally offensive and wanted to sort of test the limits of the theory saying that this you know that there's this excessive intermediation but of course I'm all joking aside the issues there were really very interesting but also seemed like they called for just a bit of you know we felt motivated to understand them further and so in this paper our contribution is to really change the model of the the good market compared to what's in that paper in that paper there's a good market which uses a salam circle demand configuration and here we're going to in a sense make things more basic by just looking at a monopoly seller rather than competition among sellers but we're also going to relax assumptions on demand curvature and the sort of punchline will be that under low convexity demand price coherence can be surprisingly good for consumers and for total welfare it's even possible that the platform lacks sufficient incentive to require price coherence and these results stem from what strikes as an interesting effect that we call the drawing in effect whereby price coherence spurs purchases spurs more purchases by consumers who have low valuations for the good which is kind of an interesting theme because if you listen to some of the discussion about if you listen to some of the discussion about say American Express or other platforms that impose this kind of restraint there's a sense that their imposition of the constraint benefits relatively wealthy people or high valuation consumers at the expense of low valuation consumers and while we're not trying to push this too far there is something kind of interesting to think about here about what's driving this and that it's really the low valuation consumers who are being drawn into the market when you have price coherence imposed by the platform all right so I will go to the model now we have a platform a unit mass of buyers and a single merchant and the basic setup is one where in the absence of a platform it would just be sort of the simplest most plain vanilla monopoly model that you can imagine buyers each you know buyers have unit demand for a good sold by the merchant they have valuation v from for the good that comes from some you know drawn from an interval between zero and v bar and there's a distribution evaluations g so in the absence of the platform the merchant would be a monopolist with demand given by this function q and it would we're assuming that it has zero marginal cost but now we add the platform to the mix buyers have the same valuations for the good but there are two groups of buyers a share lambda of them are just non-users who are going to behave exactly as the users would in the monopoly model they just purchase if the valuation for the good is greater than the price and PD here is denotes a little bit more formal with the notation in a second of prices here PD is the the price to buy the price that you have to pay to buy directly then there's the one minus lambda share of users and they have the choice to either purchase directly so they get V minus PD because they've paid directly they've bought directly and paid the direct price or they can purchase via the platform in which case they get a bonus B which you can think about as being a number of different things it could be for example free delivery that you get from the platform whereas you might have otherwise had to go pick it up it can pick the good up it can be a an improved ability to return the good or a warranty service so you can think about the B being a lot of different things this B is homogeneous across users whereas the Vs are everybody has an individual B an individual V but the B is homogeneous in this model and then if you buy through the platform you potentially pay a different price and PM denotes the mediated price so one important concept is just this one of net price where you have PM minus B as sort of the relevant thing if somebody is going to if a user is going to purchase through the platform the relevant comparison is between V and the net price and we compare two regimes the one price flexibility where the merchant can set price directly sorry set the direct price PD and also the mediated price PM we compare this with price coherence which is a regime where the platform has imposed that the merchant is allowed to just set one price for both direct sales and mediated sales and that's we call P hat and from the setup we can write the merchant's profits under the two regimes and this is very straightforward you know the merchant is receiving a lambda share of profits from the non-users in both regimes and then a one minus lambda share from the users in both regimes of course the prices are going to be important here in the price coherence is in some sense simpler because every user is going to be paying P hat but it's important to just you know do the right mechanics here the price that matters in terms of determining the marginal consumer for the case of mediated sale or for users is the net price because they're going to be getting they're going to be comparing you know users will be comparing their V evaluation for the good with the net price whereas for non-users the only thing they care about is just P and then also as far as the marginal cost we've assumed that there's zero marginal cost for the seller of of producing the good but there's this fee there's this transaction fee F that's set by the platform and so that's an endogenous quantity and so for all purchases made by users via the if users choose to make the purchase via the platform then the merchant incurs the marginal cost F whereas if non-users buy or if potentially if a user were to not make the purchase via the platform but instead purchase directly then the F wouldn't be wouldn't be part of the store wouldn't be a marginal cost so in the basic model the timing is as follows and then I'll relax this a bit even more that I'm going to proceed in three steps which you'll see the but to fix ideas let's think about this timing first the platform does two things it sets the fee F the per transaction fee and it also chooses whether to allow the merchant to have price flexibility or to impose the price coherence restraint um and I just just to clarify in the main setup in the main setup F is a per transaction fee but in the paper we also allow it to be an ad valorem fee so in practice you tend to see these things as proportion fees that are proportional to the revenue of the transaction rather than a per transaction fee and the results that I'm so so this setup that I'm talking about and that we have in the main text of the paper is a per transaction fee but that turns out not to be a crucial factor for the for the results and it's it's just more sort of more cumbers a lot of the calculations are more cumbersome when you do the proportional fee but we derive we derive everything in the appendix involving that when we use a using a proportional fee um okay so that's what the platform does then after what once the platform is moved the merchant chooses whether to participate on the platform um if it participates then under flexibility it sets the two prices and if it participates under coherence then it sets the one price as we've discussed then all of the buyers are going to choose whether to purchase the good um in solving this problem the users the the ones the buyers who are platform users can choose whether to make their purchases via the platform or to make them directly um all right so in in um this version of the story the um buyers status as either a user or a non-user is exogenously given and and it's captured by this lambda um and then at the end and and as in the Edelman and Wright paper um that's an endogenous choice and I'll talk about that um um but for for simplicity this setup you know just note that this setup is not making the choice of whether to become a user or not endogenous um the um one sorry Alex can I please um the platform users so here you assume that they are aware of the merchant right so the platform doesn't create uh the doesn't create exactly for for the for the brand exactly yes no so the issue of sort of search or providing more opportunities is not something that we're building into this story absolutely correct thank you um and so the the sort of one main assumption in the environment is that demand is strictly globally long log concave and so we're varying um you know so demand curvature is going to be an important thing here but it's not like a lot of papers on demand curvature where we say something like everything you thought was true under log concave demand turns out to be flipped around 180 degrees when demand is log convex that's not I mean that that's sort of the story of every other paper that I write but um in this story we're just restricting attention to um log concave demand and then the interest in a sense the interesting thing seems to be when demand is is a bit more you know a bit less convex than than the upper bound of this um whoops what did I do um all right so in the in the first bit of analysis and to sort of illustrate the main effect I'm going to be just talking about exogenous transaction fees so I'm going to be doing a comparison of the um coherence versus flexi flexibility regimes but for for now let's not even think about the optimal choice of f let's just say that there's some f um and that that's going to be held fixed across the two regimes and then we're going to relax that and think about the effect of varying the f optimum that you know of allowing the platform to vary f optimally in accordance with its choice of regime um so there's just a strictly positive transaction fee and you know some of the details that are in the paper will show that you know in order for this to make sense this f has to lie somewhere in this interval where it's greater than zero but below um and a level f upper bar which is strictly less than b and as long as uh f lies in this interval um described here the merchant would have an incentive to participate regardless of which regime the platform sets so that's so what so in this first exercise we're just going to assume that f is in this interval and then we solve the problem of participation for the merchant um and so first lemma says that given any such transaction fee that leads the merchant to participate under flexibility or coherence the following ranking holds um and so this you know first you can look at just these sort of these nominal prices this is this is very straightforward it says that if you think about the direct price and the mediated price um the mediated price is greater under flexibility the mediated price is going to be greater than the direct price why is that well there's a boost in demand that the users get from um getting that but that benefit be from using the platform that the non-users don't get from just purchasing directly so the mediated price is higher than the direct price and if price coherence gets imposed then the uniform price is going to be you know some kind of a weighted average of these two things it's because it's going to be the price that applies to both both groups and it's going to be in the middle um so that's nominal prices but then we can think about net prices so this is sort of the prices that determine the the cutoff or determine who the marginal users are and the net so that the only net price that matters is obviously for users not for the non-users the net price for users under right so this this inequality is is this first one is implied by this other one that the net price under coherence has to be lower than the net price under flexibility but also the net price under flexibility is lower than the direct price and two takeaways that we use from this lemma are first that coherence has a drawing in effect and by that what do we mean well when we think about the marginal users the ones who have the lowest valuation in all of this are the marginal users under coherence um marginal users under coherence have lower valuations for the good um than do marginal users under um flexibility and these are all buyers with lower valuations than the marginal non-user and then secondly the spread between marginal users and non-users valuations for the good is greater under coherence than under flexibility um so these properties are useful in allowing us to get to the first result so I'm going to talk about consumer surplus under the two regimes in this setting and to do so I want to introduce this so-called undershooting condition so this is just with this is just a condition that you can um that you can ask whether it is satisfied or not with any set of three exogenous prices and so we say that um you know so so this is the um p hat is the price is a price that might arise under price coherence PD is a direct price and PM is a um mediated price and so undershooting holds when under coherence the merchant's nominal price weekly undershoots the population weighted average of the direct and mediated prices it charges under flexibility so right so you look at the um you you look at flexibility the prices that prevail under flexibility um and then you take a population weighted average of those to compare it to the um price that arises under price coherence if that ladder price is lower than the population weighted average then we say that undershooting the undershooting condition is satisfied um and so the first proposition just involves exogenous pricing this is just for any set of prices um and net prices ranked as in the lemma if this undershooting condition is satisfied then consumer surplus is greater under coherence than it is under flexibility and that's um to talk about this let's let's look at this graph and in a way this graph has a lot of the intuition that is I think nice from from the paper and that that drives um drives the results that I that I think are new so I'll talk for a bit about this um you can start off by thinking about a benchmark where you have uh and this benchmark comes from the paper in the rand paper of Chen and Schwartz where they look at differential pricing versus uniform pricing in a context where a seller has two different markets with two different marginal costs so you can think about a seller that simply faces a single demand curve the same demand curve in both of two markets but um it uh has low marginal cost in one market and high marginal cost in another and then in that setting you compare consumer surplus under differential pricing versus uniform pricing and the the basic effect you would get there is that when you impose a uniform pricing constraint you get a reduction in the high price and an increase in the low price and this tends to harm consumer surplus because the increase of the low price um is excluding is cutting off consumer surplus that's that's given by this area a and the decrease in the high price is just creating this new consumer surplus that's represented by the area b and you know the reason for this comes from the ranking of the marginal consumers that when you're adding people who are relatively high valuation this this stems from the convexity of the consumer surplus function in effect when you're adding consumers with high valuations and taking away consumers with low valuations this will tend to decrease consumer surplus however in our setting we're we need to take into account the net prices and the net prices for the users here are the p minus b these are the platform users who are getting some benefit from using platforms so the crucial thing is that um the you know the sort of idiosyncratic feature that causes users to decide whether to buy or not is their v but in terms of users ranking compared to non-users the b is going to essentially shift them from being these high value that the marginal users are instead of being relatively high valuation users they're going to be relatively low valuation users compared to the non-users and so when you go from flexibility to coherence um instead of bringing in new people whose valuations add up to consumer surplus of area b you get you know this this taller area b plus c and so you know the the point is that the gain from this the gain in consumer surplus from this imposition is is is a positive one that the overall net effect on consumer surplus is a positive one because of the the propensity to attract the low valuations consumers rather than the high valuation consumers so i hope i've made that you know as clear as i can in a short period of time so now i'll go to the more endogenous result and to do this we will focus on the particular form of demand constant pass-through demand so this gamma is a parameter that when it's uh um raider makes demand um less and less concave um and when gamma is equal to one we just have uh linear demand um so proposition two with endogenous pricing under constant pass-through demand the merchant chooses prices that satisfy the undershooting condition if and only if gamma is less than or equal to one therefore if gamma is less than or equal to one consumers coherence gives rise to greater consumer surplus than flexibility so this in particular says that for the case of linear demand consumer surplus is greater under coherence than flexibility and also note that the linear demand tightly satisfies the undershooting condition but this undershooting condition is stronger than necessary to guarantee that consumer surplus is higher under coherence so for analytical purposes the undershooting condition is useful but it's not necessary for the result that coherence gives rise to higher consumer surplus than flexibility um and here's a graph that talks a bit more about this this shows a bit more of what I was just saying about the relationship between the undershooting condition and then the just the consumer surplus comparison so here the key thing is on the x-axis that's the gamma parameter measuring convexity one gamma equals one is the case of linear demand so that's the board that's the border between the under to the left of for values of gamma less than one undershooting is satisfied for gamma is greater than one undershooting is not satisfied but for the entire blue area so roughly speaking for values of gamma that are two or lower in this example consumer surplus is greater under coherence um now um the next step that I'm going to be wrapping up soon so I'll just sort of walk you through the the um the the sort of next steps to make this a bit you know my my emphasis has has been to try to um focus on the new effects and how I'm going to kind of make things more endogenous the next step is to just endogenize the the fees f so that there's a regime specific f rather than just holding the f fixed across the two the two regimes and the the bottom line here is that endogenizing the f to make it regime specific is going to help the case for coherence um so to talk about this we first think about a condition that we call primacy of the good um and this just says that the valuation for the good this just basically says that the platform benefit is not too big the valuation for the good of the marginal buyer in the absence of a platform um inflated by the local pass-through rate so that's something that has to be less than less than one under log concave demand um exceeds the benefit that the platform brings exceeds uh versus buying directly that that's bigger than b so you know is the this condition basically says we're not in an environment where the bs are so important relative to the um marginal consumer's valuation for the good itself um and then proposition three says that if this if condition two holds then the platform's transaction fee under coherence is less than the transaction fee under flexibility um and um so i then can give a similar type of result for linear demand um the following statements are true the transaction fee that the platform sets under coherence is strictly lower than the one it sets under flexibility consumer surplus is greater under coherence than under flexibility and in the unique sub game perfect equilibrium of the game the platform chooses to impose coherence and um moreover total surplus is greater under coherence um extending that from linear demand to the case of constant pass-through demand um we get a graph that looks similar to the one i just showed but here this graph is okay so the first thing to note um the borderline between condition one being satisfied or not is no longer a um it it moves up right the undershooting condition is easier to satisfy now because of the fact that the transaction fee is lower under coherence than it is under flexibility and so that makes it more likely that undershooting will be satisfied um and the the one thing sort of the in a sense um the the one thing that can go wrong here is that the when demand is too concave the platform will have an incentive to um choose flex not to impose coherence to choose flexibility even though the um even though imposing coherence would be better for um total surplus and consumer surplus um so now finally the the the last thing to do is to endogenize user participation um and so the setup here just follows the Edelman and Wright setup where each buyer can in can choose whether to become a user or not but signing up to for the platform causes the user to incur adjoining cost c so here we just get this timing where okay the platform is still moving first setting the fee f and choosing the flexibility choosing the regime but now in the second stage there's the simultaneous moves where um buyers are observing their value of c and deciding whether or not to join the platform and at the same time the merchant is deciding whether to join the platform and setting prices um and Alex what do I I should probably wrap up in in like two minutes or what would you say a bit more you've got four minutes thank you okay so you know I don't I don't want to get into too much detail given the the time constraint we have a way of you know we have a way of parameterizing the distribution of joining costs so that we have we sort of mix a uniform distribution of joining costs between those with zero cost and those with some cost c upper bar that's prohibitively high so that they would never join the platform and um the point is we have a mass you know the important thing is we have a mass of people with joining costs that are zero and then another mass of people with joining costs that are laid out over this interval so some consumers are potentially marginal or potentially you know it would be unclear whether they would join the platform or not whereas other users with zero joining costs are definitely joining the platform um and so I'm just going to give an example here under linear demand with specific parameter values um but this you know the the specific parameter values it's not like we chose these it's not like choosing them in a very special way is important for what I'm going to say um the main thing is just not to keep not to let the be the benefit get too high relative to sort of the the valuations for the good when that's true for all of these parameterizations of um joining costs the platform and at equilibrium the platform chooses to impose price coherence and it sets f equal to this upper bound um f bar more joy more buyers join under coherence than under flexibility so this is a key sort of um similarity to the result the theme of the Edelman and Wright paper the marginal the the the cost the joining cost of the marginal buyer under coherence is greater than the joining cost of the marginal buyer under flexibility um and total surplus and and the consumer surplus derived from purchasing the good are greater under coherence and under flexibility um but regarding um the total consumer surplus including consumer surplus from transaction costs and from joining costs that's what is a bit more ambiguous it requires that there be enough um zero joining cost users relative to these interior users in order to make it so that um net consumer surplus is higher under coherence so to wrap up I just will mention a couple of policy considerations this last point raises a new trade-off that I don't think is known or present in the certainly not from the Edelman Wright paper about the um comparison between transaction surplus as a benefit from coherence versus um excessive joining so coherence may at the same time um lead to greater transaction surplus while it also um can lead to accept to this higher joining costs at equilibrium so um these two things should be weighed against one another I mean that our model is saying that it's important to weigh these things against one another in this context and I think a um regarding policy interventions a particular thing to think about is the timing of policy interventions if you think about some policy that might potentially ban price coherence or you know put restraints on the ability of a platform to to to do this kind of price coherence um you have to be careful about doing it in a setting where the where the users have primarily already joined because in a sense if you're in a mature industry where users have already joined and then you come in and um impose this kind of uh a policy that gets rid of price coherence there's the pretend you know our model says that suggests that there's the potential that you would be sort of trying to close the the barn door after the horse has already gotten out in the sense that the joining costs may already have been incurred and the you could further um damage things by reducing the consumer surplus so um you know without being said this is obviously an abstract model and we're doing this in the context you know in a very specific context but I hope that this has been um you know that this has been useful for um shedding light on these these underlying effects so to conclude in this paper we model optional intermediaries and address the incentives and the effects of price coherence restraints we find that demand curvature plays a crucial role and in low convexity environments price coherence tends to boost consumer surplus and total surplus um when demand is sufficiently concave the platform may inefficiently prefer not to impose coherence and the reason why coherence helps is due to the drawing in effect the coherence restraint stimulates purchases by platform users with relatively low valuations for the good so with that I'll finish um and thank you very much thank you Alex for this very clear presentation and now we have a discussant um Julian Wright well thanks uh yeah it was a nice presentation and um I really enjoyed reading this paper um you know the the idea that price coherence could be actually really good for consumers is one that naturally is an interesting possibility to explore and of course other papers in the literature have found trade-offs between um you know the sort of multiple effects and it could go either way but none as clear and simple as the mechanism that you've put forward which is you know it makes it um you know nice to discuss um so what I thought I would talk about is sort of first of all the connection to our paper and I think more broadly than our QGE paper the earlier literature on credit cards and uh no surcharge rules which is you know basically equivalent to price coherence or price parity clauses uh and you know Alex you identified one key difference which is in those models we're looking at competing merchants with unit demand I think that's not just in our paper but in some of the earlier papers like rachanter rolls and so on um where they had hoteling model whereas in your paper you have monopoly seller facing elastic demand right downward sloping demand and so you have this new drawing in effect which um seems quite plausible uh which we didn't have and of course you know if you think about these markets where platforms are you know imposing price coherence I guess there is a question which is a more natural way of modeling it is it a monopoly seller facing elastic demand that has the ability to price to different buyers based on demand um or is it more that prices are determined by competition and just passing through the fees that they face so I mean that's obviously an empirical question but um in some sense what I thought you're capturing with the monopoly seller which can price discriminate um is that price discrimination by monopoly seller is bad for consumers under you know certain demand conditions and I think those demand conditions are satisfied in your in your setting so to some extent you're sort of capturing that idea that you know price discrimination is bad and we want to have we don't want to have monopolist price discriminating we want to have them setting uniform prices that helps consumers um now um aside from that difference which is you know obviously competing sellers versus monopoly seller I think there's another difference between um this earlier literature and what you're doing which you didn't sort of draw out so much which is in our paper we allow the benefit B to be in sort of endogenously determined by the platform's investment in the earlier payments literature they allowed there to be a rebate or a fee charged to the buyer side which the platform determined and so typically you know they might set a high seller fee and they might set a rebate to consumers and so natural question is what happens in your framework if you allow for that kind of endogeneity of two sided pricing basically right because I'm just trying to connect what you do in that earlier literature and that seems to me the only other key difference the other point I wanted to make so this is sort of going now a bit beyond that earlier literature in our 2015 paper is like in that paper and as in your paper the platform never charges more than be the transaction benefit that it provides right and that of course makes it harder for price coherence to be bad and you know that was a constraint we had to work under in our earlier paper and that's why I had this sort of cost mechanism with excessive intermediation but of course if you think more broadly about these platforms that provide a discoverability for the sellers right so they're providing you know cheaper search for buyers to find sellers then there's other reasons why the platform may be able to charge beyond this transaction benefit because if you know if the seller doesn't join they're not discovered and they're willing to pay more than the transaction benefit to be discovered and that's sort of the work that I did with Cheng Zi Wang in the ran paper and other work where you actually see a much stronger sort of negative effect of price parity on consumers because the platform can actually go well well beyond the transaction benefits and prevent the natural showrooming that would happen in that case by imposing price parity so I think that you know that's the sort of one related point to that sort of my last point and coming back to your model bit you know if you think about it in terms of there being two demands right demands of the non-users and demands of the users the demands of the users have higher willingness to pay because they get B from going through the platform so my question would be what happens if you have the reverse situation which is actually the demand function for the non-users has higher willingness to pay would the sort of conclusions reverse and the reason I asked that is because if you think about the direct consumers they may not be as price sensitive as the consumers that are coming to the platform like if you think about hotel bookings right the direct consumers may be not the ones that are searching as much they may not be as price sensitive they may actually have a different demand function than the ones coming through the platform and so you know had some idea that the willingness to pay of the direct consumers may be higher and more generally that captures the idea that the direct consumers may be more captive to the sellers and therefore they have a chance to charge them higher prices so you know that that seems like it might have some interesting effects in your setting and I better stop here in the interest of time very nice paper thank you thanks Julian Alex do you want to take a couple minutes to answer Julian's points and in the meantime to the participants feel free to answer questions in the chat well thank you Julian I think that those are all very good questions and some of them we've thought about some of them we've you know some of them we've thought more about some of them we haven't thought as much about and you know I don't feel as though it's important to go one by one between them because I do think that they're all very good questions and you know I certainly don't intend the thrust of this presentation or this paper to be um proselytizing that price coherence is good under all circumstances or that you know we don't need to worry about it because I think there are a lot of reasons why we do need to be worried about it and you know the intent here is very much to just clarify our thinking and understand this force and how it fits in rather than say anything sort of more ambitious you know overly ambitious more ambitious than that the showrooming is another factor that is very important in this context and you know Julian and others have have have done very interesting work on that also I think that the one of the things that we worked on earlier in this research didn't actually involve investment in the the bees so in the it's indeed true that in the Edelman and Wright paper there's this endogenous investment by the platform in creating the bee and there there's something interesting there that's not in this paper but that you know maybe we'll we'll get around to writing up at some point it sort of ended up being something that was a bit peripheral to the main point of this paper where you can also have if you have heterogeneity in the bees and you have the investment there's a potential for a kind of spence distortion there where the platform focuses on the bee of the marginal a certain type of marginal user whereas the social planner would care about the bee of the infromarginal users which is as far as I know I don't think that's in the QGE paper I could be wrong about that but that's another you know the this is just to say that there are a lot of interesting issues here that I you know I hope that we get a chance to address either in this paper future versions of this paper or in in subsequent work and that you know the goal here is just to be clearheaded and identify this note this you know this particular aspect of the relevant mechanics so I think I'll leave it at that