EXTERNALITIES AND PRICING ON MULTIDIMENSIONAL. MATCHING PLATFORMS: Online Appendix

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1 EXTERNALITIES AND PRICING ON MULTIDIMENSIONAL MATCHING PLATFORMS: Online Appendix Melati Nungsari Davidson College Sam Flanders UNC-Chapel Hill Job Market Paper September 21, 2015 Abstract We study multidimensional matching on a two-sided platform operated by a profit-maximizing monopolist. Utility is assumed to be non-transferable. Agents on one side of the market have preferences over both homogeneous and heterogenous dimensions. Homogenous traits can be interpreted as vertical characteristics, where all agents agree on the ranking of the characteristic. In contrast, agents have different preferences over heterogeneous traits. We find two previously unstudied externalities that result from our matching framework. We call the first the rivalry externality and the second the no-transfers externality. In particular, from a social welfare-maximizing perspective, we find that in a model with multidimensional matching, agents tend too match too aggressively on traits that are homogeneous and not aggressive enough on the heterogeneous dimensions. We show that these externalities are not unique to search models, but also appear in frictionless markets, such as those studied by Gale and Shapley [1962]. These two externalities have interesting pricing implications for the monopolist, who may find it optimal to charge distortionary per-interaction prices to correct for them. JEL classification: D21, C78, D83 Keywords: Externalities, Pricing, Monopolist, Matching, Search, Two-Sided Platforms We would like to thank Gary Biglaiser, Peter Norman, Fei Li, Andy Yates, Helen Tauchen, and Sergio Parreiras at the Economics Department at the University of North Carolina at Chapel Hill for their invaluable mentoring and help on this research project. We would also like to thank the participants of the UNC Micro Theory Workshop for their comments that have helped shaped this paper. Melati would also like to thank the Department of Economics at Davidson College for their support. Last but not least, we both would like to thank the Royster Society of Fellows and the Economics Department at UNC-Chapel Hill for generously funding our years working on this project in graduate school. All mistakes are our own. menungsari@davidson.edu; Economics Department at Davidson College, Box 7123, Davidson, NC 28035; (office phone); (cellphone); (fax). sflander@live.unc.edu; Economics Department at the University of North Carolina at Chapel Hill, 107 Gardner Hall, CB #3305, Chapel Hill, NC 27599; (phone); (cellphone); (fax). 1

2 Firm Operates Only One Platform - Unobservable Types Suppose now that types are unobservable to the monopolist. Even though there are many interesting questions to be asked when we consider a world where agents can lie about their type, this particular model setup is not the best one to study such issues. This is primarily due to the fact that a lie about one s type does not affect the person s ability to obtain a date since in my model, every agent receives a random draw of a date in each period that they are on the platform. In the unobservable types setting as studied here, agents can also lie to the monopolist but not to other agents on the platform, who will be able to see their type once they are on a date. This is hard to rationalize, since one would imagine that an agent should not be able to lie to the monopolist but not to other agents, particularly in an online dating platform world where characteristics are partially observable through dating profiles. We arguea that a better setup to studying lying on dating platforms would expand on the Burdett and Coles [1997] model. In any case, some of the results from observable types carry on to the case where types are unobservable. This is due to the fact that the set of pricing strategies that the monopolist can choose from decreases with the addition of constraints in the unobservable type optimization problem; thus, the pricing strategy that is optimal in the larger set of mechanisms is surely optimal in the constrained set. Conducting examples and simulations in the case where types are unobservable is significantly more complicated than when types were observable. This is due to the fact that the addition of the incentive compatibility constraints add another layer of complexity to solving for a steady-state equilibrium. For example, we could easily solve for equilibrium in the case where the distribution function for ψ was a two-point mass distribution when types were observable because of the fact that there was full surplus extraction from the point of view of the monopolist. However, this may no longer be true in the case where types are unobservable. In particular, full surplus is extracted from L types but this may not be true for H types. In the following subsections, we first highlight the monopolist s problem and then study some of the results that carry over from the case when types are observable. Monopolist s Problem Let Π θ be the profit that the monopolist obtains from agents of type θ. In equilibrium, the monopolist s beliefs about the proportion of studs on the platform is correct, and is exactly equal to α. Thus, when types are unobservable, the monopolist solves the following optimization problem: 2

3 max Π 2{αΠ H + (1 α)π L } f H,f L,p H,p L s. t. IR L : C L (p L ) f L 0 IR H : C H (p H ) f H 0 IC L : C L (p L ) f L C L (p H ) f H IC H : C H (p H ) f H C H (p L ) f L Note now that since the monopolist does not observe types, it does not have the ability to choose the proportion of any type that joins in the platform in each platform, i H and i L. Consistent with standard mechanism design results, we obtain the following proposition. Proposition 1 The monopolist optimally sets p = (p H, p L ) and f = (fh, f L ) so that IR L and IC H binds: C L (p L) = f L (1) C H (p H) C H (p L) = f H f L (2) Proof. Recall the monopolist optimization problem: max Π 2{αΠ H + (1 α)π L } f H,f L,p H,p L s. t. IR L : C L (p L ) f L 0 IR H : C H (p H ) f H 0 IC L : C L (p L ) f L C L (p H ) f H IC H : C H (p H ) f H C H (p L ) f L 3

4 We would now like to show that the IR L and IC H constraints bind. We first show that IR L has to bind. Suppose, for a contradiction, that IR L does not bind with equality. By IC H, I have C H (p H ) f H C H (p L ) f L C L (p L ) f L > 0 whereby the last inequality holds since C H (p) C L (p) for any price p in a steady-state equilibrium. Thus, IR H also does not bind with equality. This implies that the monopolist can increase f H and f L by a small amount and all constraints will still hold. This is a profitable deviation which cannot occur in equilibrium; hence, the contradiction. Now, we show that IC H must bind with equality. Suppose not, for a contradiction. Thus, C H (p H ) f H > C H (p L ) f L C L (p L ) f L = 0 This implies that the monopolist can strictly profit by increase f H by a small amount without violating any of the constraints; thus, the contradiction. That is, there is full surplus extraction from L types and the incentive compatibility constraint is set so that H types have no incentive to lie and misreport their type. Optimal Pricing There are two main results that carry over from the observable type case. These two have to do with when the shocks ψ are small, where a sufficient but not necessary condition for this is when θ L m result characterizes equilibrium per-interaction prices. δ 1 δ. The first Proposition 2 Suppose ψ [0, m] and types are unobservable by the monopolist. For the parameterizations such that θ L m δ 1 δ, the profit-maximizing monopolist optimally charges p H, p L all agents to leave in the first period that they are on the platform. Proof. sufficiently high to induce The proof for this proposition follows the proof from the results obtained in the main body of the paper, noting that the optimization problem when types are unobservable has strictly more constraints than the 4

5 optimization problem when types are observable. This being, the number of contracts that the monopolist can offer to the agents is strictly fewer under unobservable types than under observable types. In fact, the set of contracts under unobservable types is a strict subset of the set of contracts under observable types. This being, since the contract offered under the case where types are perfectly observable is optimal in the larger set of contracts and is also a member of the smaller set of contracts (unobservable types) since it does not differentiate between types, it is necessarily also the solution within the set of smaller contracts. Given that all agents leave in the first period, as we did for the case when types are observable, we can also fully characterize the optimal price schedule for both studs and duds. The next result informs us that the monopolist will set two price schedules for the types, where the fixed prices are the same and the perinteraction prices induce agents to marry and exit the platform in the first period. Note that even though the per-interaction prices for both of the types may be the same, they do not necessarily have to be. The only restriction is that they are sufficiently high. However, as pointed out for the case when types are observable, since agents do not actually pay the per-interaction prices if they marry in the first period, the contract only specifies a fixed fee that does not discriminate between types. The per-interaction prices are only set to induce agents to marry the first agent they go out on a date with. Proposition 3 Suppose θ L m δ 1 δ and types are unobservable. In the steady-state equilibrium, the profitmaximizing monopolist sets sufficiently high per-interaction prices p H, p L so that all agents marry and exit in the first period and f H = f L. Proof. This proof follows from the results obtained in the main body of the paper and the reasoning from Proposition 2. References Armstrong, M. (2006). Competition in Two-Sided Markets. RAND Journal of Economics 37(3), Adachi, H. (2003). A Search Model of Two-Sided Matching under Nontransferable Utility. Journal of Economic Theory 113(2),

6 Abdulkadiroğlu, A., P. Pathak, and A. Roth. (2005). The New York City High School Match. American Economic Review Papers and Proceedings (May 95), Abdulkadiroğlu, A., P. Pathak, A. Roth, and T. Sönmez. (2005). The Boston Public School Match. American Economic Review Papers and Proceedings (May 95), Azevedo, E. and J. Leshno. (2012). A Supply and Demand Framework for Two-Sided Matching Markets. Working Paper. Becker, G. S. (1973). A Theory of Marriage: Part 1. Journal of Political Economy 81(4), Bester, H. (1994). Price Commitment in Search Markets. Journal of Economic Behavior and Organization 25(1), Bloch, F. and H. Ryder. (2000). Two-Sided Search, Marriages, and Matchmakers. International Economic Review 41(1), Browning, M., P. Chiappori, and Y. Weiss. (2014). Economics of the Family. Cambridge Surveys of Economic Literature: Cambridge University Press. Bulow, J. and J. Levin. (2006). Matching and Price Competition. American Economic Review, 96(3), Burdett, K. (1978). A Theory of Employee Job Search and Quit Rates. American Economic Review, 68(1), Burdett, K. and M. Coles. (1997). Marriage and Class. Quarterly Journal of Economics 112, Burdett, K. and R. Wright. (1998). Two-sided Search with Nontransferable Utility. Review of Economic Dynamics 1(1), Cabral, L. (2011). Dynamic Price Competition with Network Effects. Review of Economic Studies 78, Cornelius, T. (2003). A Search Model of Marriage and Divorce. Review of Economic Dynamics 6(1), Damiano, E. and L. Hao. (2007). Price Discrimination and Efficient Matching. Economic Theory 30, Flanders, S. (2014). Mirror Matches: Analyzing Matching Markets with N-Dimensional Preferences. Working Paper. Gale, D. and L.S. Shapley. (1962). College Admissions and the Stability of Marriage. The American Mathematical Monthly 69(1),

7 Hagiu, A. (2006). Pricing and Commitment by Two-Sided Platforms. RAND Journal of Economics 37(3), Halaburda, H. and M. Piskorski. (2011). Competing by Restricting Choice: The Case of Search Platforms. Harvard Business School Working Paper Hoppe, H., B. Moldavanu, and A. Sela. (2011). The Theory of Assortative Matching Based on Costly Signals. Economic Theory 47, Hotelling, H. (1929). Stability in Competition. The Economic Journal 39(153), Kojima, F. and P. Pathak. (2009). Incentives and Stability in Large Two-Sided Matching Markets. American Economic Review 99(3), McCall, J. (1970). Economics of Information and Job Search. The Quarterly Journal of Economics 84(1), Mortensen, D. and C. Pissarides. (1994). Job Creation and Job Destruction in the Theory of Unemployment. Review of Economic Studies 61, Rochet, J. and J. Tirole. (2006). Two-Sided Markets: A Progress Report. RAND Journal of Economics 37(3), Rochet, J. and J. Tirole. (2010). Platform Competition in Two-Sided Markets. Journal of the European Economic Association 1(4), Rogerson, R., R. Shimer, and R. Wright. (2005). Search-Theoretic Models of the Labor Market: A Survey. Journal of Economic Literature Vol XLIII, Roth, A., T. Sönmez, and M.U. Ünver. (2005). Pairwise Kidney Exchange. Journal of Economic Theory 125, Roth, A., T. Sönmez, and M.U. Ünver. (2007). Efficient Kidney Exchange: Coincidence of Wants in Markets with Compatibility-Based Preferences. The American Economic Review 97(3), Rysman, M. (2009). The Economics of Two-Sided Markets. Journal of Economic Perspectives 23(3), Salop, S. C. (1979). Monopolistic Competition with Outside Goods. The Bell Journal of Economics 10(1),

8 Shapley, L.S. and M. Shubik. (1971). The Assignment Game I: The Core. International Journal of Game Theory 1(1), Shimer, R. and L. Smith. (2001). Matching, Search, and Heterogeneity. Advances in Macroeconomics 1(1). Weyl, G. (2010). A Price Theory of Multi-Sided Platforms. American Economic Review 100(4),