Enabling Versus Controlling
Andrei Hagiu Julian Wright

Working Paper
16-002 July 7, 2015

Copyright © 2015 by Andrei Hagiu and Julian Wright Working papers are in draft form. This working paper is distributed for purposes of comment and discussion only. It may not be reproduced without permission of the copyright holder. Copies of working papers are available from the author.

Enabling versus controlling
Andrei Hagiu∗ and Julian Wright† July 7, 2015

Abstract In an increasing number of industries, ﬁrms choose how much control to give professionals over the provision of their services to clients. We study the tradeoﬀs that arise in choosing between a traditional mode (where the ﬁrm takes control of service provision) and a platform mode (where professionals retain control over service provision). The choice of mode is determined by the need to balance two-sided moral hazard problems arising from investments that only professionals can make and investments that only the ﬁrm can make, while at the same time minimizing distortions in decisions that either party could make (e.g. promotion and marketing of professionals’ services, price setting, choice of service oﬀering, etc). JEL classiﬁcation: D4, L1, L5 Keywords: platforms, theory of the ﬁrm, vertical integration, control rights, moral hazard

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Introduction

In many service industries (e.g. consulting, education, home services, legal, outsourcing, staﬃng, taxi), traditional ﬁrms employ service providers (professionals) and control most (if not all) of the relevant decision rights pertaining to the interactions with customers—work schedule and scope, service fees, and how services are marketed and performed. The last decade has brought a rapidly increasing number of ﬁrms in these industries that take advantage of information technologies to enable independent professionals to connect directly with customers, such that professionals control some or all of the relevant decisions (e.g. Coursera, Elance-oDesk, Gerson Lehrman Group, Lyft and Uber, Task Rabbit, etc.). Motivated by these examples, this paper studies a ﬁrm’s choice between two modes of organization—a traditional mode versus a platform mode—where the key diﬀerence between the
∗ †

Harvard Business School, Boston, MA 02163, E-mail: ahagiu@hbs.edu Department of Economics, National University of Singapore, Singapore 117570, E-mail: jwright@nus.edu.sg

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two modes is that professionals hold more control rights in the platform mode than in the traditional mode.1 The model we develop captures that both professionals and the ﬁrm typically add value to the services oﬀered to clients. Speciﬁcally, each makes a costly and non-transferable investment (or eﬀort) decision which raises the revenue that can be obtained from clients. This creates a two-sided moral hazard problem for the ﬁrm. However, two-sided moral hazard is not suﬃcient to generate an interesting tradeoﬀ between the two modes given that we allow the ﬁrm to compensate professionals based on the revenues obtained from clients. To create an interesting tradeoﬀ we add a third decision variable that can either be controlled by the ﬁrm or the professional, i.e. a transferable decision variable. The allocation of control rights over this third transferable decision variable is what determines the mode of organization in our model. If control rights are given to professionals then the ﬁrm operates in the platform mode. If control rights are instead kept by the ﬁrm then it operates in the traditional mode. We show that, in the presence of two-sided moral hazard, a meaningful tradeoﬀ exists between the platform mode and the traditional mode only if the transferable decision variable is non-contractible, and is either costly (e.g. promotional activities, investments in equipment, etc.) or exhibits spillovers across multiple professionals (e.g. prices, horizontal marketing decisions). In this setting, we ﬁrst show that the optimal contract involves a linear (two-part) contract, with a ﬁxed payment and a ﬁxed portion of revenue being paid between the two parties. Since revenues need to be split between the two parties, in general both professionals’ and the ﬁrm’s non-transferable eﬀorts will fall short of the ﬁrst-best level, as will the transferable decision variable if it is costly. In the baseline model without spillovers across professionals, or interaction eﬀects between the three decision variables, we show that the party whose moral hazard problem is more important should receive a greater share of the revenue. This implies the same party should also be given control over the transferable decision variable to lessen the distortion from the transferable decision variable being set too low. Thus, we predict the traditional mode is chosen when the ﬁrm’s moral hazard problem is more important and the platform mode is chosen when the professionals’ moral hazard problem is more important. The tradeoﬀ is more complex when there are spillovers across professionals’ transferable decisions. Consider ﬁrst the case when the transferable decision is a revenue-increasing, costly investment (e.g. marketing or equipment). If a larger investment by one professional also increases the revenue obtained by other professionals providing services through the same ﬁrm (i.e. the spillover is positive), then an increase in the magnitude of the spillover always shifts the tradeoﬀ between the two modes in favor of the traditional mode, as expected. This is because in the traditional mode, the ﬁrm coordinates investment decisions to internalize the spillover, whereas in the platform mode, individual professionals leave it uninternalized and therefore invest too little. Furthermore, the baseline result, according to which the traditional (respectively, platform) mode is more likely to be chosen when the ﬁrm’s (respectively, the professionals’) moral hazard becomes more important, continues to hold. Things
This focus is also consistent with legal deﬁnitions that emphasize control rights as the most important factor determining whether professionals providing services through a given ﬁrm should be considered independent contractors (platform mode) or employees (traditional mode).
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are more interesting with negative spillovers. In platform mode, individual professionals now invest too much by not internalizing the spillovers. But these higher investments can help oﬀset the primary distortion due to revenue sharing, namely that the party with control rights invests too little because it keeps less than 100% of the revenue generated. The platform mode can then be a useful way for the ﬁrm to get professionals to choose higher levels of the transferable decision variable without giving them an excessively high share of revenues. This eﬀect has two counter-intuitive consequences. First, when negative spillovers are not too large in magnitude, an increase in their magnitude shifts the tradeoﬀ in favor of the platform mode, the opposite of the usual case. Second, if the magnitude of negative spillovers is suﬃciently large, then professionals get a lower share of revenues in the platform mode than in the traditional mode. This leads to a reversal of the baseline logic, according to which control rights over the transferable decision variable are more likely to be given to the ﬁrm (respectively, the professionals) when the magnitude of the ﬁrm’s (respectively, the professionals’) moral hazard problem increases. In the case when the transferable decision variable is the price for the professionals’ services, the tradeoﬀ between the two modes is determined by diﬀerent considerations. Since setting a higher price does not involve any real cost, revenue-sharing does not distort price-setting in either mode, so revenue-sharing can be used to balance the two-sided moral hazard problem equally well in both modes. However, a higher price raises the return to each party from costly investments, thereby mitigating each party’s moral hazard problem. When services are substitutes, independent professionals set prices too low in platform mode, which therefore exacerbates moral hazard. As a result, the traditional mode dominates. On the other hand, if professionals’ services are complements, then independent professionals set prices too high in the platform mode, thus mitigating each of the moral hazard problems. As a result, we ﬁnd that the platform mode can be preferred. The next section discusses related literature. Section 3 provides some examples of markets in which the traditional mode versus platform mode choice that we model is relevant. Section 4 introduces our theory and obtains results for the simpler case with a single professional, while Section 5 extends the theory to the case with multiple professionals and spillovers. Section 7 concludes.

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Related literature

A key and novel contribution of our paper is to extend in a natural way the theory of the ﬁrm based on control rights and incentive systems (see Grossman and Hart, 1986, Hart and Moore, 1990) to platforms. Our focus on platforms makes our work quite diﬀerent from earlier works studying the classic “make” versus “buy” decision. The “make” part is similar: a ﬁrm operating in the traditional mode controls most decisions but still needs to design contracts in order to address moral hazard by employees. However, the “enabling” (platform) scenario is quite diﬀerent from “buying” (i.e. contracting via the market). The diﬀerence is that the “platform” mode gives professionals control rights over the transaction with end-customers. In a “buy” relationship between ﬁrm and suppliers, the ﬁrm still has complete control over the ﬁnal payoﬀs that arise from selling the ﬁnal good or service

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to customers (this diﬀerence is discussed in detail in our previous paper, Hagiu and Wright 2015b). We focus on ex-post moral hazard, hence the need to provide incentives in the form of revenue sharing. We show that linear contracts remain optimal in our setting despite the fact that both the principal and the agent take non-contractible actions after the contract is signed (two-sided moral hazard) and one of the parties takes a third payoﬀ-relevant action (the transferable decision variable) that also depends on the contract signed. In this respect, we extend the earlier literature showing the optimality of linear contracts (see Holmstrom and Milgrom, 1987 and especially Romano, 1994). This paper relates to two of our earlier works that study how ﬁrms choose to position themselves closer to or further from a multi-sided platform business model. The focus on incentive systems and moral hazard in the current paper contrasts with Hagiu and Wright (2015a), which applied the “adaptation theory of the ﬁrm” to marketplaces.2 The adaptation theory emphasized the advantage of a marketplace over a reseller in allowing third-party suppliers to adapt decisions to their local information. We abstract from information advantages in the present theory. Closer to the current paper is Hagiu and Wright (2015b), which provides some initial analysis of the choice a ﬁrm faces between operating in the traditional way or being a multi-sided platform. A key diﬀerence is that in Hagiu and Wright (2015b) we only allowed for one-sided moral hazard. In the current paper, regardless of which party controls the choice of the transferable variable, the other party still makes non-contractible decisions that aﬀect the outcome. This feature of our model captures what we think is a key characteristic of platforms: even though a platform enables professionals to interact with customers on terms they control, the platform still makes important decisions that aﬀect the revenues derived by professionals. Thus, in contrast to our earlier work, here we introduce two-sided moral hazard, which is fundamental to the tradeoﬀs we study. Another diﬀerence is that the current model is much more general, and applies to a wider range of ﬁrms rather than just multi-sided platforms facing cross-group network eﬀects. In the literature on multi-sided platforms, a few other authors have noted the possibility that platforms can sometimes choose whether or not to vertically integrate into one of their sides, although they have not modelled this choice: Gawer and Cusumano (2002), Evans et al. (2006), Gawer and Henderson (2007) and Rysman (2009). The tradeoﬀs these works discuss revolve around platform quality and product variety and therefore are quite diﬀerent from the ones we identify here.

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Examples

The most natural examples of markets in which the choice we study is relevant involve ﬁrms that can either employ professionals and control how they deliver services to clients, or operate as platforms enabling independent professionals to provide services directly to clients. While this choice has become particularly prominent due to the proliferation of Internet-based services (e.g. Coursera, Elance-oDesk, Handy and Homejoy, Hourly Nerd, Lyft and Uber, Task Rabbit, etc.), it has been long relevant in a number of “oﬄine” industries.
2

See Gibbons (2005) for a classiﬁcation of diﬀerent theories of the ﬁrm.

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The hair salon industry is a good example as it has long featured both modes of organization. Some salons employ their hair stylists and pay them ﬁxed hourly wages plus commissions that are a percentage of sales. Such salons control schedule and product lines, provide all the necessary equipment, do all the marketing to customers and provide stylists with some training and guidance. In contrast, other salons rent out chairs (booths) to independent hair stylists. The stylists keep all earnings minus booth rental fees paid to the salon: these fees are usually some combination of ﬂat weekly fees and a variable percentage of sales. In such salons, each stylist decides her/his own schedule and product line, provides her/his own equipment and advertises to customers. The salon owners still make all necessary investments to maintain the facilities, as well as to advertise the salon to customers. Another oﬄine example that may be more familiar to readers is economic consulting ﬁrms, such as Analysis Group, Charles River Associates, Cornerstone Research, and National Economics Research Associates (NERA). Almost all of these ﬁrms use a hybrid between the two modes of organization, relying both on in-house consultants that are employed, and outside economists that act as independent professionals. The latter set their own work schedule and fees (the ﬁrms typically add a percentage fee on top and charge the total to clients). There is signiﬁcant variation across ﬁrms in the share of in-house versus independent consultants. For instance, NERA relies mainly on in-house consultants, whereas Cornerstone Research relies mainly on independent consultants. Our model can also potentially ﬁt intermediaries for products rather than just services, provided either (i) the price charged by intermediaries to buyers is contracted upon between suppliers and the intermediary (e.g. resale price maintenance clauses) or (ii) the contracts between suppliers and intermediaries specify variable fees as percentage of revenues generated (or of the price to buyers) instead of nominal wholesale prices.3 Some product intermediaries act as platforms (marketplaces) and others as traditional resellers that take control over most decisions relevant to the sale to buyers (see Hagiu and Wright, 2013 and 2015a). As an example, consider two online intermediaries facilitating the sale of used cars. eBay Motors is a pure marketplace where sellers retain all relevant control rights regarding their listing (price, photos, information, warranties and other ancillary services oﬀered, etc). eBay charges car sellers a ﬁxed listing fee as well as a fee that depends on the sale price of the car. By contrast, Beepi facilitates the sale of used cars from individuals to other individuals, but takes control over many relevant aspects of the sale. Beepi sends an inspector to each seller to appraise the car and suggest a price, guarantees sale within 30 days at that price (if not, Beepi buys the car itself), takes professional and standardized pictures with Beepi plates on, arranges delivery to the buyer and oﬀers buyers a range of ancillary services (10-day money back guarantee, complete warranty for several months). Beepi adds 3-9% on top of the price suggested to the seller and charges the total to the buyer. Table 1 shows how these and other examples where ﬁrms may choose between the two modes ﬁt our theory. In particular, it illustrates how the revenue generated by each professional (or supplier) can depend on each of the three diﬀerent types of non-contractible decision variables featured in our
The reason for these conditions is that in our model we assume only total revenue is contractible, not the underlying demand. Either one of these two conditions ensures that contracting on revenue is equivalent to contracting on demand.
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Table 1: Examples
Transferable decisions Non-transferable investment decisions made by professionals eﬀort regarding service quality and/or customer experience (friendliness, before and after service, etc.) customer service (e.g. friendliness, politeness); investment in local information investments in development of skills and expertise; eﬀort put into understanding and responding to customer needs; eﬀort supplied in the provision of the service investments in development of skills and expertise; customer service; eﬀort supplied in the provision of the service quality of content and its delivery; eﬀort put into understanding and responding to students needs quality and/or maintenance of the product Non-transferable investment decisions made by the ﬁrm investments in maintenance of the salon (cleanliness, signage), common washing and coloring facilities and advertising of the salon to customers; training and guidance provided to professionals advertising the service (company) to customers; investments in the quality of the corresponding app and back-end infrastructure (e.g. payment processing, dispatch system) investments in the (online) infrastructure that allows communication and monitoring by the client; investment in payment functionality; advertising the ﬁrm to corporate clients

Hair salons

investments in equipment and uniforms; advertising of individual professionals’ services to customers (online and oﬄine) quality, maintenance and cleanliness of car subject to minimal requirements; work schedule advertising of individual professionals services; work schedule and scope

Uber & Lyft vs. traditional taxi companies

Elance-oDesk vs. traditional staﬃng and outsourcing agencies (e.g. Adecco, Infosys) Hospitals & their clinics

equipment; support staﬀ; work schedule and scope; advertising of individual clinics services curriculum design (topics, length, assessment); advertising of individual instructors and courses advertising and presentation of individual cars; after-sale customer service and guarantees

Coursera vs. University of Phoenix eBay Motors vs. Beepi

development and maintenance of common infrastructure (e.g. physical space, common staﬀ, any shared equipment); advertising the hospital (e.g. website) investments in online infrastructure for content delivery, interactions, feedback and evaluations (both ways); advertising the brand investments in the website and related infrastructure (e.g. payment system, fulﬁllment, delivery, customer service, etc.); advertising of the website to users

model: (i) a costly investment always chosen by the ﬁrm; (ii) a costly investment always chosen by the professional; and (iii) a transferable decision that is chosen by the ﬁrm in traditional mode and by the professional in platform mode. We have not included the price (or fee) charged to customers in Table 1, which is potentially another transferable decision variable in each of the examples listed. However, the price is also sometimes contractible or pinned down by market constraints, in which case it can be treated as a ﬁxed constant in our analysis. Finally, while we have included an example in Table 1 in which ﬁrms act as intermediaries for the sale of products, for conciseness we will focus on the case of professional services in the rest of the paper.

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4

General model with one professional

We start by considering a model with a single professional. Section 5 will allow for multiple professionals and consider spillovers between them.

4.1

Assumptions

There is a ﬁrm (the principal) and a professional (the agent). The revenue generated by the ﬁrm and the professional is R (a, e, E), which depends on three types of actions, all of which are noncontractible. Actions e and E are non-transferable: the ﬁrm always chooses E ∈ R+ at cost cE (E) and the professional always chooses e ∈ R+ at cost ce (e). This means there is two-sided moral hazard. To ﬁx ideas, one can think of E as capturing the ﬁrm’s ongoing investments in infrastructure, and of e as the eﬀort made by the professional in the provision of the service. Action a is transferable, i.e. it can be chosen either by the ﬁrm or by the professional, depending on the mode in which the ﬁrm chooses to operate. The party that chooses a ∈ R+ incurs cost ca (a). Our analysis encompasses two possibilities: • Costly actions which always increase revenues, i.e. ca (a) > 0 for a > 0 and R increasing in a. Examples include marketing or promotional activities, customer service eﬀorts, investments in equipment, etc. • Costless actions (ca = 0), such that R is single-peaked in a. Price is the most natural example, but such actions also include “horizontal choices” (see Hagiu and Wright 2015a), e.g. the allocation of a ﬁxed promotional capacity between emphasizing the professional’s previous education and work experience versus her/his performance on recent projects through the ﬁrm. We assume throughout the paper that the only variable that can be contracted on is the realized revenue R(a, e, E). In other words, any contract oﬀered by the ﬁrm to the professional can only depend on R(a, e, E), but not on any of the underlying variables (a, e, E). We make the following technical assumptions4 : (a1) All functions are twice continuously diﬀerentiable in all arguments. (a2) The cost functions ce and cE are increasing and strictly convex in their arguments. If ca = 0 then ca is also increasing and strictly convex. Furthermore, ca (0) = ca (0) = ce (0) = ce (0) = cE (0) = cE (0) = 0. a e E (a3) The revenue function R is non-negative for all (a, e, E), strictly increasing and weakly concave in (e, E). If a is costless (i.e. if ca = 0), then R is concave and single-peaked in a for all (e, E). If a is costly (i.e. if ca = 0), then R is strictly increasing and weakly concave in a.
Subscripts indicate derivatives throughout the paper. Thus, ca indicates the derivative of ca with respect to a, and a Ra indicates the partial derivative of R with respect to a.
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(a4) lime→∞ (Re (a, e, E) − ce (e) < 0) for all (a, E) and limE→∞ RE (a, e, E) − cE (E) < 0 for all e E (a, e). If ca = 0 then for all (e, E) there exists a (e, E) such that R (a, e, E) = 0 for all a ≥ a (e, E). If ca = 0 then lima→∞ (Ra (a, e, E) − ca (a)) < 0 for all (e, E). a (a5) For all t ∈ [0, 1], each of the following two systems of three equations in (a, e, E) admits a solution:   tRa (a, e, E) = ca (a) a  (1 − t) Re (a, e, E) = ce (e) e   tRE (a, e, E) = cE (E) E and   (1 − t) Ra (a, e, E) = ca (a) a  e (e) (1 − t) Re (a, e, E) = ce   tRE (a, e, E) = cE (E) . E These assumptions are standard and are made to ensure that the optimization problems considered below are well-behaved. Assumption (a4) ensures there is always a ﬁnite solution to the optimization problems we consider. The ﬁrst set of equations in (a5) are the ﬁrst-order conditions corresponding to the traditional mode, while the second set of equations in (a5) are the ﬁrst-order conditions corresponding to the platform mode.5 Note that if R (a, e, E) is additively separable in its three arguments then (a5) is implied by (a1)-(a4) and the solution to each of the two sets of equations is unique for all t ∈ [0, 1]. The ﬁrm has all the bargaining power and we denote the professional’s outside option by W0 . The ﬁrm can choose to operate in one of two modes: T -mode (traditional) and P -mode (platform). In both modes, the ﬁrm oﬀers the professional a contract consisting of a ﬁxed fee F and a variable fee tR (a, e, E), where t ∈ [0, 1]. This means the net payment from the professional to the ﬁrm is F + tR(a, e, E), and the professional is left with (1 − t)R(a, e, E) − F . In the next subsection, we show that the restriction to such linear contracts is without loss of generality. The diﬀerence between the two modes is that in T -mode, the ﬁrm controls the transferable action a, whereas in P -mode a is chosen by the professional. This generally implies diﬀerent levels of R(a, e, E) across the two modes, and diﬀerent optimal contracts (t, F ). Thus, it is possible for F to be negative under T -mode (i.e. the professional receives a ﬁxed wage) and positive under P -mode (i.e. the professional pays a ﬁxed fee). Nevertheless, if W0 is high enough then the professional will receive a net payment in both modes. Note also that in our model it is immaterial whether the ﬁrm or the professional collects revenues R (a, e, E) and pays the other party their share. If in T -mode the ﬁrm collects revenues and pays (1 − t)R(a, e, E) to the professional then this can be interpreted as a bonus in an employment relationship. Given that the ﬁrm holds all the bargaining power, it will set F in both modes so that the professional is indiﬀerent between participation and her outside option, which for convenience we normalize throughout to W0 = 0.
A simple suﬃcient condition for (a5) to hold is that there exist a, e, E such that R (a, e, E)−ca (a)−ce (e)−cE (E) < 0 whenever a > a, e > e or E > E. Indeed, this condition ensures that the relevant space in (a, e, E) is compact, so we can apply the Kakutani ﬁxed point theorem for existence of the solutions to the two systems of equations.
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The game we study has the following timing. In stage 0, the ﬁrm chooses whether to operate in T -mode or P -mode. In stage 1, the ﬁrm sets (t, F ) and the professional decides whether to accept and pay the ﬁxed fee F . In stage 2, there are two possibilities depending on the ﬁrm’s choice in stage 0. In T -mode, the ﬁrm chooses E and a, and the professional simultaneously chooses e. In P -mode, the ﬁrm chooses E and the professional simultaneously chooses e and a. Finally, in stage 3, revenues R (a, e, E) are realized; the ﬁrm receives tR(a, e, E) and the professional receives (1 − t)R(a, e, E).

4.2

General results

We ﬁrst establish that the restriction to linear contracts in both modes is without loss of generality (the proof is in the appendix). Proposition 1 In both modes, the ﬁrm can achieve the best possible outcome with a linear contract.

This proposition implies that the ﬁrm’s proﬁts in T -mode can be written as6 ΠT ∗ = max R (a, e, E) − ca (a) − ce (e) − cE (E) (1)

t,a,e,E

subject to   tRa (a, e, E) = ca (a) a  (1 − t) Re (a, e, E) = ce (e) e   tRE (a, e, E) = cE (E) . E Similarly, the ﬁrm’s P -mode proﬁts are ΠP ∗ = max R (a, e, E) − ca (a) − ce (e) − cE (E)

(2)

t,a,e,E

(3)

subject to   (1 − t) Ra (a, e, E) = ca (a) a  (1 − t) Re (a, e, E) = ce (e) e   E (E) . tRE (a, e, E) = cE

(4)

Assumption (a5) ensures the existence of a solution (a, e, E) to (2) and to (4) for any t ∈ [0, 1]. If there are multiple solutions for a given t then the way we have written the optimization programs above implicitly assumes that the ﬁrm can choose a stage 2 Nash equilibrium that maximizes its proﬁts. Note that, in general, the respective proﬁts yielded by both modes are lower than the ﬁrst-best proﬁt level max R (a, e, E) − ca (a) − ce (e) − cE (E) .
a,e,E

At the optimum, the ﬁxed fee F of the linear contract is always set such that the participation constraint of the professional is binding, i.e. (1 − t) R (a, e, E) − F − ce (e) = 0.

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The reason is two-sided moral hazard: the payoﬀ R (a, e, E) needs to be divided between the ﬁrm and the professional in order to incentivize each of them to choose their respective actions. In other words, this ineﬃciency is the moral hazard in teams identiﬁed by Holmstrom (1982), where a team here consists of the professional and the ﬁrm. To reach the eﬃcient solution, Holmstrom (1982) shows that one needs to break the budget constraint, i.e. credibly commit to “throw away” revenue in case a target speciﬁed ex-ante is not reached. This type of solution is unrealistic in the contexts we have in mind. Furthermore, our focus is not on oﬀering general solutions to this class of problems, but rather to analyze the tradeoﬀs between the two modes of organization, both of which are unable to reach the ﬁrst-best. Comparison of programs (1) and (3) makes it clear that the diﬀerence between the two modes comes from the choice of the non-transferable action a. The tradeoﬀ between the T -mode and the P -mode boils down to whether it is better to align the choice of a with the ﬁrm’s choice of eﬀort E (T -mode) or with the professional’s choice of eﬀort e (P -mode).

Proposition 2 Compare the ﬁrm’s proﬁts under the two modes. (a) If the transferable action a is contractible or costless (i.e. ca = 0), then the two modes are equivalent and lead to the same ﬁrm proﬁts (ΠT ∗ = ΠP ∗ ). (b) Suppose the transferable action a is non-contractible and costly. If the non-transferable action e is contractible or if it has no impact on revenue (Re = 0) then ΠT ∗ > ΠP ∗ . If the non-transferable action E is contractible or if it has no impact on revenue (RE = 0) then ΠP ∗ > ΠT ∗ . Proof. For (a), if a is contractible then the constraint in a disappears in both modes, so the programs (1) and (3) become identical. If ca = 0, then the constraint in a is the same in both modes and is deﬁned by Ra (a, e, E) = 0, so the two modes are equivalent once again. For (b), if the professional’s eﬀort has no impact on revenues (Re = 0) then professionals will set e = 0 in both modes. In T -mode it is then optimal for the ﬁrm to retain the entire revenue (t = 1), so proﬁts are ΠT ∗ = max R (a, 0, E) − ca (a) − cE (E) .
a,E

This is clearly higher than proﬁts under P -mode: ΠP ∗ = max R (a, 0, E) − ca (a) − cE (E)
t,a,E

subject to (1 − t) Ra (a, 0, E) = ca (a) a tRE (a, 0, E) − cE (E) . E If the professional’s eﬀort e is contractible then in T -mode the ﬁrm optimally sets t = 1 and proﬁts are ΠT ∗ = max R (a, e, E) − ca (a) − ce (e) − cE (E) ,
a,e,E

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i.e. the ﬁrst-best level of proﬁts, which strictly dominate the proﬁts that can be achieved in P -mode. By a symmetric argument, we obtain the result for the case when the ﬁrm’s eﬀort has no impact on revenues (RE = 0) or E is contractible. Thus, for there to exist a meaningful tradeoﬀ between the two modes with a single professional, (i) all three actions must be non-contractible and have a strictly positive impact on revenues R, and (ii) the non-transferable action a must carry a strictly increasing cost ca (a). In particular, the addition of any number of contractible transferable actions will not have any meaningful impact on the tradeoﬀ. Furthermore, part (a) of the proposition implies that if the transferable action a is price then, even if it cannot be contracted on, the two modes are equivalent. As we will see in section 5.4, this no longer holds when there are multiple professionals and the price for each individual service generates spillovers on the revenues generated by other professionals. In the general case of interest, when all three actions are non-contractible, have a positive impact on revenues and carry strictly increasing costs, the two modes distort the choice of a, but they do so in diﬀerent ways, leading to diﬀerent proﬁts. Heuristically, if the ﬁrm’s moral hazard (E) is more important (in the sense that it has a larger impact on R) then the optimal t is higher in both modes, but then the T -mode induces relatively less distortion in a and is therefore more likely to be preferred. Conversely, if the professional’s moral hazard (e) is more important then the optimal t is lower in both modes, so that the P -mode induces relatively less distortion in a and is therefore more likely to be preferred.

4.3

Additively separable case

To achieve a better understanding of the tradeoﬀ between the T -mode and P -mode in the case of a single professional, in this section we assume the revenue function is fully additively separable, i.e. R (a, e, E) ≡ ra (a) + re (e) + rE (E) , where ra , re and rE are strictly increasing and weakly concave, and ca , ce and cE are strictly increasing and convex. The expression of R (a, e, E) could result from additively separable demand with an exogenously ﬁxed price for the professional’s service. Deﬁne (a (t) , e (t) , E (t)) as the respective solutions to the following equations:
a tra (a) = ca (a) a e tre (e) = ce (e) e E trE (E) = cE (E) . E

Assumptions (a1)-(a4) imply that a (t), e (t) and E (t) are well-deﬁned, unique and increasing in t. Furthermore, a (0) = e (0) = E (0) = 0. With this notation, when the ﬁrm sets a linear contract with variable fee t ∈ [0, 1] in stage 1, the stage 2 equilibrium actions are (a (t) , e (1 − t) , E (t))) in T -mode and (a (1 − t) , e (1 − t) , E (t))) in P -mode.

11

Denote π a (t) ≡ ra (a (t)) − ca (a (t)) π e (t) ≡ re (e (t)) − ce (e (t)) π E (t) ≡ rE (E (t)) − cE (E (t)) . It is easily veriﬁed that under assumptions (a1)-(a4), the functions π a (t), π e (t) and π E (t) are all increasing in t.7 Resulting proﬁts for the two modes are then ΠT ∗ = max π a (t) + π e (1 − t) + π E (t)
t

(5) (6)

ΠP ∗ = max π a (1 − t) + π e (1 − t) + π E (t)
t

These two expressions show that the diﬀerence between the two modes lies in the distortion created by each party not keeping all of the revenue attributable to the transferable action. In T -mode, a higher variable fee means the ﬁrm keeps a higher share of the revenue generated by the transferable action, and there is less distortion in the proﬁts π a (t) generated by the transferable action. In P mode, a higher variable fee means the professional keeps a lower share of the revenue generated by the transferable action, so there is more distortion in the proﬁts π a (1 − t) generated by the transferable action. Denote also tT ∗ ≡ arg maxt π a (t) + π e (1 − t) + π E (t) tP ∗ ≡ arg maxt π a (1 − t) + π e (1 − t) + π E (t) the respective, optimal variable fees charged by the ﬁrm in the two modes. Proposition 3 The ﬁrm keeps a larger share of revenue in T -mode than in P -mode, i.e. tT ∗ ≥ tP ∗ . Furthermore, if tT ∗ < 1/2 then ΠP ∗ > ΠT ∗ ; if tP ∗ > 1/2 then ΠP ∗ < ΠT ∗ . Proof. Since π a , π e and π E are increasing functions, we have tT ∗ ≥ arg max π e (1 − t) + π E (t) ≥ tP ∗ .
t

For the second part of the proposition, since π a (t) is increasing, the following inequality holds for all t < 1/2: π a (1 − t) + π e (1 − t) + π E (t) > π a (t) + π e (1 − t) + π E (t) . Thus, if tT ∗ < 1/2 then ΠT ∗ < π a 1 − tT ∗ + π e 1 − tT ∗ + π E tT ∗ ≤ π a 1 − tP ∗ + π e 1 − tP ∗ + π E tP ∗ = ΠP ∗ , and symmetrically for tP ∗ > 1/2. The ﬁrst part of the proposition conﬁrms the common intuition according to which independent contractors working through platforms should claim a larger share of the revenues that they directly
7 a a For example, πa (t) = (1 − t) ra (a (t)) at (t) > 0.

12

generate (i.e. a larger commission) than employees working for ﬁrms. This is because in P -mode, sharing revenues with the ﬁrm leads to a higher distortion of the transferable variable and lower proﬁt; by contrast, in T -mode, the more revenue the ﬁrm keeps, the lower the distortion of the transferable variable and the higher the proﬁt. We will see in Section 5.3 that this is no longer always true with N > 1 professionals and spillovers. The second part of the proposition can be interpreted as stating that a ﬁrm would never ﬁnd it optimal to function in T -mode and pay bonuses above 50% or function in P -mode and charge variable fees above 50%. In other words, according to this prediction of our model, if the professional receives or keeps more than 50% of variable revenues, the ﬁrm is functioning in P -mode, and not in T -mode. This prediction is supported by our examples. Traditional hair salons that employ their hair stylists oﬀer bonuses ranging from 35% to 60% of sales, whereas the variable fee charged by salons that rent chairs (when such a fee is used) ranges from 30% to 40% of independent stylists’ sales, thus leaving 60%-70% for the stylists. Task Rabbit retains a total of approximately 35% of the total amount paid by clients to taskers, who act as independent contractors.8 oDesk retains approximately 9% of the amount paid by employers to its independent contractors.9 Finally, Lyft and Uber retain 20% of the price per ride paid by users, which is consistent with the P -mode (although the two companies control price, their drivers are free to choose the cars they drive and their work schedule).

4.4

Additively separable example

Suppose the revenue generated by the service provided by the professional is R (a, e, E) = θa + γe + δE, where θ, γ and δ are all positive constants. The ﬁxed costs are assumed to be 1 1 ca (a) = a2 , ce (e) = e2 2 2 1 and cE (E) = E 2 . 2

Thus, γ can be interpreted as the importance of professionals’ moral hazard, whereas δ represents the importance of the ﬁrm’s moral hazard. Relegating calculations to an online appendix available from the authors’ websites, we obtain tT ∗ = tP ∗ = and the following proposition. Proposition 4 The ﬁrm prefers the P -mode to the T -mode if and only if γ 2 > δ 2 .
8 Task Rabbit adds to the hourly rate set by its taskers a service fee set such that it is equal to 30% of the total paid by the client (e.g. $30 if the tasker rate is $70), plus a trust and safety fee equal to 5% of the total. 9 oDesk adds a 10% fee on top of the rate charged by the contractor.

θ2 + δ 2 + γ 2 + δ2 δ2 θ2 + γ 2 + δ 2 θ2

13

In other words, the ﬁrm prefers the P -mode if professionals’ moral hazard is more important than the ﬁrm’s moral hazard. In particular, in this example the tradeoﬀ does not depend on θ, the impact of the transferable action on revenues. The reason is that in both modes the share of revenues retained by the party that chooses the transferable action (tT ∗ in T -mode and 1 − tP ∗ in P -mode) is increasing in θ. Since tT ∗ and 1 − tP ∗ increase at the same rate in this particular example (due to the symmetry of T -mode and P -mode proﬁts in δ 2 and γ 2 ), the resulting tradeoﬀ does not depend on θ.

5

General model with multiple professionals and spillovers

In this section, we extend the model from Section 4 to N > 1 identical professionals and introduce the possibility that the non-transferable action ai can also impact the revenue generated by each of the other professionals j = i (i.e. that there are spillovers).

5.1

Assumptions

To keep the analysis as streamlined as possible, we assume that revenue is large enough relative to costs such that it is optimal for the ﬁrm to induce all N professionals to join in both modes. Then the revenue attributable to professional i who joins the ﬁrm (in P -mode or T -mode) when all N professionals join is R (ai , si , ei , E), where si ≡ σ (a−i ) and σ is a symmetric function of the transferable actions chosen by the professionals (or for the professionals) that join other than i, with values in R+ . In the speciﬁc examples used below, σ will be the average of these other actions, i.e. σ (a−i ) = For convenience, we denote by → ≡ (a, ..., a) − an
n j=i aj

N −1

.

the vector made of n coordinates all equal to a, and by σa (a) the partial derivative of σ (a−i ) with − respect to any j = i, evaluated at →N −1 (symmetry implies that all these partial derivatives are a equal). As before, ei is the non-transferable action (eﬀort) chosen by professional i and E is the nontransferable action (investment) chosen by the ﬁrm. Note that the ﬁrm chooses a single E that impacts the revenues attributable to all N professionals. Also note that R (ai , si , ei , E) does not depend on the choices of non-transferable actions ej for other professionals j = i. As we discuss below, introducing this possibility would not add any meaningful aspects to the tradeoﬀ between the two modes that we focus on.

14

The costs of the transferable and non-transferable actions are the same as before and the same across professionals: ca (ai ), ce (ei ) and cE (E). Finally, the ﬁrm is not allowed to price discriminate, i.e. it is restricted to oﬀer the same contract Φ (R) to all professionals. The technical assumptions (a1)-(a2) from section 4 remain as before. Assumptions (a3)-(a5) are adapted as follows: (a3’) The revenue function R (a, s, e, E) is non-negative for all (a, s, e, E), strictly increasing and weakly concave in (e, E). If ca = 0 then R (a, s, e, E) is concave and single-peaked in a for all (s, e, E) and ca
N i=1 R (ai , σ (a−i ) , ei , E)

is concave and single-peaked in all ai for all (ei , E) and i ∈ {1, .., N }. If
N i=1 R (ai , σ (a−i ) , ei , E)

= 0 then R (a, s, e, E) is strictly increasing and weakly concave in a and

is strictly increasing and weakly concave in all ai , i ∈ {1, .., N }. (a4’) lime→∞ (Re (a, s, e, E) − ce (e)) < 0 for all (a, s, E) and limE→∞ RE (a, s, e, E) − cE (E) < e E 0 for all (a, s, e). If a is price then for all (s, e, E) there exists a (s, e, E) such that R (a, s, e, E) = 0 for all a ≥ a (s, e, E). If a is not price then lima→∞ (Ra (a, s, e, E) − ca (a)) < 0 for all (s, e, E). a (a5’) For all t ∈ [0, 1], each of the following two systems of three equations in (a, e, E) admits a solution:  − − a a  t (Ra (a, σ (→N −1 ) , e, E) + (N − 1) σa (a) Rs (a, σ (→N −1 ) , e, E)) = ca (a) a  → − e (e) (1 − t) Re (a, σ ( a N −1 ) , e, E) = ce   − tN RE (a, σ (→N −1 ) , e, E) = cE (E) a E and  − a  (1 − t) Ra (a, σ (→N −1 ) , e, E) = ca (a) a  → − (1 − t) Re (a, σ ( a N −1 ) , e, E) = ce (e) e   → − E (E) . tN R (a, σ ( a ) , e, E) = c
E N −1 E

(a6’) The optimization problem solved by the ﬁrm admits a well-deﬁned solution which is symmetric in all N professionals in both modes. The main addition to (a3) is to ensure that the spillover is not so large that it overcomes the “main” eﬀect of ai . Assumption (a6’) is an additional assumption, which is used to rule out asymmetries in the optimal solution due to spillovers. It is always satisﬁed in the absence of spillovers. The timing is the same as in Section 4.

5.2

General results

We ﬁrst establish the analogous result to Proposition 1 (the proof is in the appendix). Proposition 5 If assumptions (a1)-(a2) and (a3’)-(a6’) hold, then in both modes the ﬁrm can achieve the best possible symmetric outcome with a linear contract.

15

The proposition implies that the ﬁrm’s proﬁts in T -mode can be written − ΠT ∗ = max N (R (a, σ (→N −1 ) , e, E) − ca (a) − ce (e)) − cE (E) a
t,a,e,E

(7)

subject to  − − a a  t (Ra (a, σ (→N −1 ) , e, E) + (N − 1) σa (a) Rs (a, σ (→N −1 ) , e, E)) = ca (a) a  → − e (e) (1 − t) Re (a, σ ( a N −1 ) , e, E) = ce   − tN R (a, σ (→ a ) , e, E) = cE (E)
E N −1 E

(8)

Similarly, the ﬁrm’s proﬁts in P -mode can be written ΠP ∗ = − max N (R (a, σ (→N −1 ) , e, E) − ca (a) − ce (e)) − cE (E) a (9)

t,a,e,E

subject to  − a  (1 − t) Ra (a, σ (→N −1 ) , e, E) = ca (a) a  → − (1 − t) Re (a, σ ( a N −1 ) , e, E) = ce (e) e   − tN R (a, σ (→ a ) , e, E) = cE (E) .
E N −1 E

(10)

Comparing the two programs above, there are now two diﬀerences between the two modes, both originating in the choice of the non-transferable actions ai . The ﬁrst diﬀerence is the same as in the case N = 1: the ﬁrst-order condition in a has a factor t in T -mode and a factor (1 − t) in P -mode. The second diﬀerence is new and stems from the presence of spillovers across the N professionals: in T -mode the ﬁrm internalizes the spillover when setting ai for i = 1, .., N , whereas the spillovers are left uninternalized in P -mode when each ai is chosen by individual professional i. We can now derive the corresponding proposition to Proposition 2. Proposition 6 Compare the ﬁrm’s proﬁts under the two modes. (a) If the transferable actions ai are contractible, then the two modes are equivalent and lead to the same ﬁrm proﬁts (ΠT ∗ = ΠP ∗ ). If the transferable actions are costless and non-contractible (i.e. ca = 0), then the two modes lead to diﬀerent proﬁts except when there are no spillovers (Rs = 0). If in addition the revenue function is additively separable in (a, s), e and E (i.e. if it can be written R (ai , si , ei , E) = ras (ai , si ) + re (ei ) + rE (E)) then ΠT ∗ > ΠP ∗ . (b) Suppose the transferable actions are non-contractible. If the non-transferable actions ei are contractible or if they have no impact on revenue (Re = 0), then ΠT ∗ > ΠP ∗ . If ca = 0 and the non-transferable action E is contractible or has no impact on revenue (RE = 0), then ΠT ∗ > ΠP ∗ . Proof. For part (a), if ai is contractible then the ﬁrst constraint in (8) and the ﬁrst constraint in (10) disappear, so the programs (7) and (9) become identical. If the actions ai carry no cost (ca = 0) then these ﬁrst constraints remain distinct in the two modes, unless Rs = 0. Suppose in addition that R (a, s, e, E) is additively separable. Then in stage 2, the equilibrium choices of (e, E) as functions of t are identical in both modes. Denote them by (e (t) , E (t)). The ﬁrm’s T -mode proﬁts can then be 16

written: − max N ras (a, σ (→N −1 )) + N (re (e (t)) − ce (e (t))) + N rE (E (t)) − cE (E (t)) a
t,a

− − as as subject to ra (a, σ (→N −1 )) + (N − 1) σa (a) rs (a, σ (→N −1 )) = 0 a a which is equal to − max N ras (a, σ (→N −1 )) + N (re (e (t)) − ce (e (t))) + N rE (E (t)) − cE (E (t)) . a
t,a

This is strictly higher than P -mode proﬁts − max N ras (a, σ (→N −1 )) + N (re (e (t)) − ce (e (t))) + N rE (E (t)) − cE (E (t)) a
t,a

− as subject to ra (a, σ (→N −1 )) = 0. a For part (b), if the professionals’ eﬀorts are contractible or if Re = 0 then the ﬁrm can achieve the ﬁrst-best level of proﬁts in T -mode by setting t = 1, obtaining − ΠT ∗ = max N (R (a, σ (→N −1 ) , e, E) − ca (a) − ce (e)) − cE (E) . a
a,e,E

In P -mode, we know the resulting proﬁts are strictly lower because the choice of a is not ﬁrst-best optimal (it does not account for spillovers). If ca = 0 and E is contractible or RE = 0 then the ﬁrm can once again achieve the ﬁrst-best level of proﬁts in T -mode, this time by setting tT arbitrarily close to 0, obtaining − ΠT ∗ = max N (R (a, σ (→N −1 ) , e, E) − ca (a) − ce (e)) − cE (E) . a
a,e,E

In P -mode it is also optimal to set tP arbitrarily close to 0 but resulting proﬁts are less than ﬁrstbest because the choice of a is not ﬁrst-best optimal (it does not account for spillovers). As a result, ΠT ∗ > ΠP ∗ . There are two key diﬀerences in Proposition 6 relative to Proposition 2. First, due to spillovers, the case with ca = 0 no longer leads to equivalence. This reﬂects that in T -mode, spillovers are internalized, whereas in P -mode they are not. One may think that this always leads to the T -mode to dominate the P -mode, but this is only true when the revenue function is additively separable in all its arguments or when E is contractible or when E has no impact on revenue. If instead all three types of actions are non-contractible and impact revenues and there are interaction eﬀects between a and the two types of non-transferable eﬀort, then either mode may dominate. In particular, interaction eﬀects between ai and ei or between ai and E may either exacerbate or dampen the disadvantage of the P -mode in terms of not internalizing spillovers.

17

The second diﬀerence is that in case (b), contractibility of E or RE = 0 no longer necessarily implies that the P -mode dominates. The advantage of the P -mode in achieving the constrained ﬁrstbest level of ei must still be traded-oﬀ against the advantage of the T -mode in internalizing spillovers. At the extreme, if, in addition, the transferable action is costless then the T -mode can also achieve the constrained ﬁrst-best level of ei , which implies that the T -mode does strictly better. Note that all the results in Proposition 6 would continue to hold even if we allowed for spillovers of eﬀort ei across revenues attributable to other agents j = i (accompanied by the appropriate changes in assumptions (a3’)-(a6’)). Indeed, the respective ﬁrst-order conditions corresponding to e in programs (7) and (9) would stay the same: the spillover from professionals’ eﬀorts remains uninternalized in both T -mode and P -mode because in both modes professionals choose ei ’s individually. Thus, the tradeoﬀ between the two modes would not be materially impacted by spillovers generated by the noncontractible, non-transferable eﬀorts ei . This is why we have abstracted away from such spillovers. Based on Proposition 6, the two simplest scenarios in which the tradeoﬀ between the two modes is meaningful are: 1. Costly transferable actions ai and additively separable revenue function R (ai , si , ei , E). 2. Costless transferable actions ai (namely, prices) and non-additively separable revenue function R (ai , si , ei , E). The two cases exhibit diﬀerent mechanisms—we consider them in the next two subsections through two speciﬁc examples. These two cases also correspond to realistic scenarios. In many contexts prices are easily observable and contracted on, which means they do not have an impact on the T -mode versus P -mode distinction. Alternatively, in other cases parties cannot observe price or quantity separately, so can only contract on revenue. Then price becomes a relevant transferable and non-contractible variable.

5.3

Additively separable example with spillovers

This section extends the example studied in section 4.4 to the case of N > 1 professionals and spillovers. Speciﬁcally, the revenue generated by the service provided by professional i is R (ai , si , ei , E) = θai + x (si − ai ) + γei + δE and si = σ (a−i ) is the average of the transferable actions chosen for services j = i: σ (a−i ) = We can therefore write directly R (ai , a−i , ei , E) = θai + x (a−i − ai ) + γei + δE. 18
j=i aj

N −1

≡ a−i .

Thus, when spillovers are negative (x < 0), revenue R is decreasing in a−i , which means that in P -mode the transferable actions ai are set too high. Conversely, when spillovers are positive (x > 0), revenue R is increasing in a−i , so that in P -mode the ai ’s are set too low. For example, if ai represents advertising then negative (respectively, positive) spillovers occur when services are substitutes (respectively, complements). We assume x<θ and x (θ − x) < N δ 2 , (11)

which ensures that (i) assumptions (a1)-(a2) and (a3’)-(a6’) are satisﬁed for this example, and (ii) the optimal variable fee is strictly between 0 and 1 in both modes. Note that all x < 0 are permissible under (11). We obtain (all calculations are in the online appendix) tT ∗ = tP ∗ = and the following proposition. Proposition 7 The ﬁrm prefers the P -mode to the T -mode if and only if − θ2 − N δ 2 − θ2 (θ2 + γ 2 + N δ 2 ) + γ 4 ≤ x γ2 ≤ −θ2 − N δ 2 + θ θ2 (θ2 + γ 2 + N δ 2 ) + γ 4 (13) θ2 + N δ 2 θ2 + γ 2 + N δ 2 N δ 2 − x (θ − x) (θ − x)2 + γ 2 + N δ 2

(12)

In other words, the P -mode is preferred if and only if the magnitude of spillovers is not too large. Note that the entire range of x deﬁned by (13) is permissible by assumptions (11) for θ suﬃciently large. Inspection of (13) reveals that the range of spillover values x for which the ﬁrm prefers the P -mode is skewed towards negative values. To understand why, recall that negative spillovers cause the ai ’s to be set too high in P -mode, which oﬀsets the primary revenue distortion (ai ’s being set too low because the party choosing ai does not receive the full marginal return when 0 < t < 1). Thus, when spillovers are negative, choosing the P -mode (and thereby letting professionals choose ai ) provides a way for the ﬁrm to commit to achieving a level of ai closer to the ﬁrst-best. By contrast, positive spillovers cause the ai ’s to be set too low in P -mode, which exacerbates the primary revenue distortion. This makes the P -mode relatively less likely to dominate. There still exists a range of positive spillovers for which the P -mode is preferred but that range is smaller than the corresponding range of negative spillovers. We now investigate the impact of γ 2 and N δ 2 on the tradeoﬀ between P -mode and T -mode, i.e. on the proﬁt diﬀerential ΠP ∗ − ΠT ∗ . From (13), this impact seems diﬃcult to ascertain. Fortunately, one can use ﬁrst-order conditions and the envelope theorem, which lead to simple conditions (see the online appendix for calculations).

19

Proposition 8 A larger γ shifts the tradeoﬀ in favor of P -mode (i.e. if tP ∗ < tT ∗ . A larger δ shifts the tradeoﬀ in favor of T -mode (i.e. tP ∗ < tT ∗ .

∂ (ΠP ∗ −ΠT ∗ ) ∂(γ 2 ) ∂ (ΠP ∗ −ΠT ∗ ) < ∂(N δ 2 )

> 0) if and only 0) if and only if

In other words, the eﬀects of both types of moral hazard on the tradeoﬀ conform to common intuition whenever the share of revenues retained by the ﬁrm is larger in T -mode. Recall from the analysis in section 4 that this is always the case in the absence of spillovers. However, with spillovers this may no longer be the case, so the eﬀects of the two types of moral hazard can be counter-intuitive. In particular, from (12) we obtain that, with spillovers, tP ∗ > tT ∗ if and only if x θ θ2 + N δ 2 + <− θ θ−x γ2 (14)

i.e. if the spillover x is suﬃciently negative (recall all x < 0 are permissible under assumptions (11)). When the inequality in (14) holds, professionals oﬀset the primary revenue-sharing distortion with the distortion due to spillovers in P -mode. As a result, a higher t induces less distortion in P -mode, so the ﬁrm can charge a higher t in P -mode to the point that tP > tT . When this occurs, professionals retain a lower share of revenues in P -mode than in T -mode, so their choice of non-transferable eﬀort ei is more distorted in P -mode. Consequently, when professionals’ eﬀort (moral hazard) becomes more important in this parameter region, the T -mode becomes relatively more attractive. Similarly, when the ﬁrm’s eﬀort (moral hazard) becomes more important in the same parameter region, the P -mode becomes relatively more attractive. This counter-intuitive scenario can never occur in the absence of spillovers in the additively separable case.
∗ ∗

5.4

Non-additively separable example (prices)

We now turn to the other case of interest identiﬁed in Section 5.2: the transferable action is the price for each professional’s service, and therefore does not carry any costs. Furthermore, in this case the revenue function is not additively separable, although the underlying demand function is. Speciﬁcally, the revenue generated by the service provided by professional i is now R pi , p−i , ei , E = pi d + θpi + x p−i − pi + γei + δE , (15)

where d > 0 is the demand intercept and p−i is the average of the prices chosen for services j = i. To ensure (a1’)-(a6’) are satisﬁed for this example, we assume θ < 0, γ > 0, δ > 0 −2θ + min {0, 2x} > max N δ 2 , γ 2 . (16)

Note that (16) implies all x > 0 are permissible and x > θ, so demand d + θpi + x p−i − pi + γei + δE is decreasing in pi . 20

From (15), positive spillovers (x > 0) correspond to the usual case with prices, i.e. substitute services. Also note that one could replace pi with qi (quantities), but then the usual case in which services are substitutes would be captured by negative spillovers (x < 0). Deﬁne k≡ 1 1 1 + 2 ∈ , +∞ . 2 Nδ γ |θ|

We then obtain (the calculations are in the online appendix): Proposition 9 The ﬁrm prefers the P -mode if and only if10 − 4 (1 + θk) < x < 0. k (1 + 2θk)

First, note that the proposition identiﬁes a meaningful tradeoﬀ since any x satisfying the last inequality above also satisﬁes (16) provided θ is suﬃciently negative, as do all positive x. Second, the T -mode is always preferred if spillovers are positive (services are substitutes) or if spillovers are very negative (services are strong complements). The logic is diﬀerent here relative to the case with costly transferable actions. Given that the transferable action here (price) does not carry any costs, there is no distortion of price in either mode due to revenue-sharing between the ﬁrm and each professional. As a result, the variable fee t can be used in both modes to balance the two-sided moral hazard problem (ei versus E) equally well. Furthermore, due to the strategic complementarity between pi and (ei , E), the choice of pi can either oﬀset or compound the two moral hazard problems. The T -mode has an advantage in internalizing spillovers across services. This explains why there is a larger region over which the T -mode dominates. But the fact that professionals do not internalize spillovers in P -mode can work in favor of the P -mode when spillovers are negative (x < 0). Namely, when x < 0, the P -mode leads to excessively high choices of pi , which can help oﬀset the twosided moral hazard problem. This is because a higher pi leads to higher ei and E due to strategic complementarity, which partially corrects the problem of ei and E being too low that arises from revenue sharing and moral hazard. In contrast, when x > 0 (positive spillovers), the P -mode leads to pi being set too low, which compounds the two-sided moral hazard problem. As a result, the T -mode always dominates in that case. Third, the parameters measuring the strength of the two moral hazard problems, N δ 2 and γ 2 , have the same eﬀect on the tradeoﬀ between the two modes (through k). This surprising result stands in contrast to the additively separable case where they work in opposing directions. The explanation is as follows. Again, since the transferable action (price) is not distorted by the variable fee t in either mode, both modes do just as well in terms of balancing the two-sided moral hazard problem. As noted above, when spillovers are negative, raising prices reduces the moral hazard problems due to the strategic complementarity between prices and eﬀorts, and this works equally well for both ei and E. Thus, the extent to which the P -mode is preferred over the T -mode when moral hazard problems become more important does not depend on the source of the moral hazard, but only on its magnitude.
10 4(1+θk) Recall θk < −1 so − k(1+2θk) < 0.

21

4(1+θk) Finally, it is easily veriﬁed that − k(1+2θk) is decreasing for k ∈

k ≥

1 |θ|

+

2 2|θ| .

√

Thus, assuming negative spillovers, when γ 2 and γ2 and γ2 and N δ2

√ 2 1 1 , |θ| + 2|θ| |θ| N δ 2 are small

and increasing for (i.e. k large), the

range of x over which the P -mode is preferred increases as vice versa when

N δ2

increase (i.e. k decreases); and

are large (k small). In other words, when the two-sided moral hazard

problem is of small importance, the eﬀectiveness of the P -mode in compensating for moral hazard with excessive prices increases as moral hazard becomes more important, so the tradeoﬀ shifts in favor of the P -mode. And vice versa when two-sided moral hazard is already very important.

6

Extensions

This section explores several extensions to our model: changing the timing of the infrastructure investment E; allowing the transferable action to not be entirely speciﬁc to the ﬁrm-professional relationship; and allowing the ﬁrm to choose a hybrid mode that lies between the pure T -mode and pure P -mode.

6.1

Timing

When E represents a basic infrastructure investment that is fundamental to the ﬁrm’s operations, it may be more natural that this investment is made prior to the choice of business model (T -mode versus P -mode), rather than afterwards. This reﬂects that it may be easier for a ﬁrm to change its business model than its basic infrastructure. In this case, the model becomes very similar to that in Hagiu and Wright (2015b), but without private information. The net result of this change in timing is to shift the ﬁrm’s business model tradeoﬀ in favor of the P -mode. Indeed, if the ﬁrm is able to commit to its choice of E prior to the choice of business model, then the need to keep a larger share of variable revenues in order to motivate investments in E disappears. This observation implies that one of the factors that determines the choice of mode is the extent to which the ﬁrm’s investment and eﬀort is determined upfront versus ongoing. Thus, when the ﬁrm’s ongoing investments in infrastructure (or other forms of common investment) are more important than its ex-ante investments, the tradeoﬀ shifts in the same way as our analysis above predicts when the ﬁrm’s moral hazard problem becomes more important.

6.2

The transferable action is not entirely relationship speciﬁc

The transferable action a can drive an additional wedge between the two modes when it is no longer entirely speciﬁc to the relationship between a professional and the ﬁrm. To see this simply, consider the model with one professional. Suppose that a inﬂuences some outside payoﬀs, Y (a) for the ﬁrm and y (a) for the professional. For example, eﬀective marketing choices for the services provided by professionals may also increase demand for some complementary products that the ﬁrm may be selling (e.g. some hair salons also sell hair products) and also the ability of professionals to sell their services elsewhere (e.g. a hair stylist may work at a salon but may also occasionally provide his or her services

22

in other venues). This results in two changes in programs (1) and (3) that determine T -mode and P -mode proﬁts respectively: • in both modes the ﬁrm is now maximizing R (a, e, E) + Y (a) − ca (a) − ce (e) − cE (E) • the respective ﬁrst-order conditions in a are now tRa (a, e, E) = ca (a) − Ya (a) a (1 − t) Ra (a, e, E) = ca (a) − ya (a) . a Consequently, it is clear that Y (a) and/or y (a) play eﬀectively the same role as the cost function ca (.), except that they introduce the possibility that the cost of a may be diﬀerent depending on whether it is chosen by the professional or by the ﬁrm. Predictably, it can be shown that a larger Ya (a) shifts the tradeoﬀ in favor of the T -mode, whereas a larger ya (a) shifts the tradeoﬀ in favor of the P -mode.

6.3

Hybrid mode across services

Hybrid modes, with some professionals oﬀering their services in T -mode and others in P -mode, are found quite often in the markets we consider (hair and nail salons, oDesk, etc). We show below that a strictly hybrid mode can be optimal even without spillovers (i.e. we assume Rs = 0) and despite the fact that all N professionals are identical. This is due to the fact that E is a common investment across all services oﬀered, for instance corresponding to an investment in a common infrastructure, and to the concavity of the proﬁt function with respect to E. We focus on the additively separable case with no spillovers, which, at ﬁrst glance, is the least likely scenario for a hybrid mode to be optimal (no interaction eﬀects and no asymmetries between ﬁrm and professionals). Suppose the ﬁrm functions in T -mode with respect to professionals i ∈ {1, .., n} and in P -mode with respect to professionals i ∈ {n + 1, .., N }, where n ≤ N . Thus, the ﬁrm oﬀers contract tT , F T to the n professionals that work in T -mode (employees) and contract tP , F P to the N − n professionals that work in P -mode (independent contractors). Using the notation in Section 4.3, the n employees each choose a level of eﬀort equal to e 1 − tT , whereas the N − n independent contractors each choose a level of eﬀort equal to e 1 − tP equal to a 1 − a tT tP . Finally, the level E tT , tP chosen by the ﬁrm solves E tT , tP = arg max ntT rE (E) + (N − n) tP rE (E) − cE (E) = E t ,
E

and a level of the transferable activity

. For the n employees, the ﬁrm chooses a level of the transferable action equal to

where t≡

n T N −n P t + t N N

is the “average” transaction fee collected by the ﬁrm. 23

The ﬁxed fees for employees and independent contractors are set to render both indiﬀerent between working for/through the ﬁrm and their outside option. Consequently, the total proﬁt of the ﬁrm is ΠH tT , tP , n = n π a tT + π e 1 − tT Note that ΠH tT , tP , n = N ΠP tP = ΠT tT + (N − n) π a 1 − tP + π e 1 − tP and ΠH tT , tP , n = 0 + N πE t . and

= ΠP tP , where ΠT tT

are the expressions of “pure mode” proﬁts (5) and (6) in Section 4.3.

It is easily seen from the expression of ΠH tT , tP , n that a necessary condition for the optimal choice of n to be interior (i.e. strictly between 0 and N ) is concavity of π E . The key reason is that the ﬁrm can only choose a single E, which aﬀects all professionals. If the ﬁrm could choose diﬀerent Ei ’s for each individual professional i, then it is easily seen that the optimal solution would be n = N or n = 0. Given the ﬁrm’s proﬁt function with respect to E is concave, the ﬁrm does better with an intermediate value of E (i.e. that arising from a mix of modes) than it would get from having all professionals in one mode or the other. To obtain closed-form solutions, consider the additively separable example from Section 4.4: R (ai , ei , E) = θai + γei + δE with quadratic costs 1 1 1 ca (ai ) = a2 ; ce (ei ) = e2 ; cE (E) = E 2 2 i 2 i 2 In this example, π E (t) is strictly concave: π E (t) = δ 2 t (2 − t) 2

The optimal number of employees is then (see the online appendix for the full derivation)          N γ 2 (θ2 +γ 2 −N δ 2 )
2N δ 2 θ2

if

N δ 2 > θ2 + γ 2
θ2 γ 2 2θ2 +γ 2

n∗ =

N

1−

if θ2 + γ 2 > N δ 2 > γ 2 − if N δ2 < γ 2 −

0

θ2 γ 2 2θ2 +γ 2

Note that n∗ is increasing in N δ 2 (the importance of the ﬁrm’s moral hazard) and decreasing in γ 2 (the importance of professionals’ moral hazard), consistent with the intuition built in section 4.4.

6.4

Hybrid modes across actions

In our model above, we have always restricted attention to a single transferable action for concision. In many real-world examples, however, there are multiple relevant transferable actions (see Table 1 in Section 3). This provides another dimension along which ﬁrms can (and oftentimes do) operate in hybrid modes, with some transferable decisions controlled by professionals and others by the ﬁrms. Our model can be extended to encompass this dimension as well. Consider the case with 1 profes-

24

sional but multiple transferable actions, aj , with j ∈ {1, .., M }. Assume the revenue function has the additively separable form
M

R a1 , .., aM , e, E =
j=1

raj aj + re (e) + rE (E) .

The ﬁxed cost associated with transferable action aj is caj aj . The costs associated with nontransferable actions are ce (e) and cE (E) as before. Similarly to the treatment of the additively separable case in section 4.3, deﬁne aj (t) ≡ arg max traj (a) − caj (a)
a

π aj (t) ≡ raj aj (t) − caj aj (t) π e (t) and π E (t) are the same as in section 4.3. With this notation, it is easily veriﬁed that the ﬁrm’s revenues when it charges variable fee t and keeps control over decisions j ∈ A ⊂ {1, .., M } while giving the professional control over decisions j ∈ {1, .., M } \A are Π (A, t) = π E (t) +
j∈A

π aj (t) +
j ∈A /

π aj (1 − t) + π e (1 − t) .

The ﬁrm optimizes this revenue expression over both A and t. First, the following proposition establishes that at least two functions π aj1 (t) and π aj2 (t) (j1 = j2 ∈ {1, ..M }) must be non-colinear in order for a hybrid interior mode to be optimal. Proposition 10 If π aj (t) = αj π a (t) + βj for some set constants (α1 , β1 , .., αM , βM ) with αi > 0 for all i, then the optimal mode is either A = ∅ or A = {1, .., M }. Proof. Denote by t∗ the optimal transaction fee corresponding to A∗ and suppose A∗ = ∅ and A∗ = {1, .., M }. Let then j1 ∈ A and j2 ∈ {1, .., M } \A. If π aj1 (t∗ ) < π aj1 (1 − t∗ ) then the ﬁrm could strictly improve proﬁts by giving up control over action j1 to the professional and keeping t∗ unchanged (proﬁts would increase by π aj1 (1 − t∗ ) − π aj1 (t∗ )), which contradicts the optimality of (A∗ , t∗ ). If π aj1 (t∗ ) > π aj1 (1 − t∗ ) then colinearity implies π aj2 (t∗ ) > π aj2 (1 − t∗ ), so the ﬁrm could strictly improve proﬁts by taking over control over action j2 from the professional and keeping t∗ unchanged, which once again contradicts optimality of (A∗ , t∗ ). Thus, either A = ∅ or A = {1, .., M }.

Perhaps surprisingly, collinearity holds for many simple functional forms:
α

• raj (a) = θj aα and caj (a) = aα+β , with α > 1 and β > 0, leads to π aj (t) = θjβ • raj (a) = θj ln a and caj (a) = a leads to π aj (t) = θj ln θj + θj (ln t − t)

+1

tα α+β

α β

1−

tα α+β

25

• raj (a) = θj a and caj (a) = eca with c > 0 leads to π aj (t) =

θj c

ln

θj c

+

θj c

(ln t − t)

In general, however, it is possible that the optimal mode is a strictly interior split of control rights over actions {1, .., M }. A simple example which does not lead to colinearity and therefore can generate a strictly interior split of control rights is raj (a) = θj a − This leads to
2 π aj (t) = θj

a2 2

and caj (a) =

a2 . 2

t 1 + θj t

1−

θj + 1 t 2 1 + θj t

Note that π aj (t) is increasing in t and θj for all θj > 0. The analysis above shows how strictly interior splits of control rights over transferable actions can be optimal even when there are no asymmetries between the ﬁrm and the professional regarding the costs of undertaking these actions or their eﬀectiveness (impact on revenues in our model). It is, however, important to recognize that in some real-world examples such asymmetries are an important factor in determining which control rights are held by the ﬁrm and which are held by professionals.11 For instance, Uber and Lyft have a clear advantage in setting prices for rides over their drivers (better information due to large amounts of data), wheareas drivers are in a better position to choose their work schedules and the amount of maintenance their individual cars need. Things are diﬀerent in the case of Airbnb. Individual apartment owners set prices because they arguably have better relevant information. On the other hand, Airbnb has ﬁgured out over time that it can be more eﬀective at marketing individual properties on the site than the properties’ owners. As a result, Airbnb now pays professional photographers to take pictures of owners’ properties and decides which pictures to post on the site (this is still an optional service for owners but many opt in). In yet another example, hair salons may take control over the choice of “uniforms” because they can obtain lower costs due to scale eﬀects (relative to individual hair stylists), but the marketing of hair stylists may be more eﬃciently done by each individual.

7

Conclusion

By substantially reducing the costs of remote monitoring and communication, the widespread adoption of Internet and mobile technologies has made it possible to build marketplaces and platforms for a rapidly increasing variety of services, ranging from house cleaning to programming, consulting and legal advice. As a result, the choice between platform mode and traditional mode, and the associated tradeoﬀs that we have examined in this paper are becoming increasingly relevant in a growing number of sectors throughout the economy.
Recall from above that such asymmetries may also reﬂect the fact that non-transferable actions are not entirely relationship-speciﬁc.
11

26

Speciﬁcally, we have considered the choice a ﬁrm faces between keeping control over transferable actions that aﬀect the revenues extracted from customers (traditional mode) and giving that control to professionals or suppliers (platform mode). At the most fundamental level, we have shown that the tradeoﬀ comes from balancing two-sided moral hazard, while at the same time minimizing distortions in the choice of the transferable action due to revenue-sharing or spillovers. Our modelling approach and some of our key results are reminiscent of the theory of the ﬁrm based on property rights and incentive systems. In particular, in the baseline model without spillovers, a key prediction is that control rights over the transferable actions should be given to whichever party’s (the ﬁrm or the professionals) moral hazard problem is more important. However, we have shown that this prediction can be over-turned when spillovers are introduced, both with costly and costless transferable actions. In particular, when the transferable action is price (and therefore there are interaction eﬀects with the non-transferable actions), the tradeoﬀ between the two modes no longer depends on the source of moral hazard, but only on its magnitude. These insights are novel and contribute to extending the theory of the ﬁrm based on property rights and incentive systems to new types of organizational choices—platforms versus traditional ﬁrms. Our analysis is also relevant to current legal and regulatory debates about whether “sharing economy” service marketplaces should be forced to treat professionals that work through them as employees rather than independent contractors.12 In particular, all existing legal deﬁnitions emphasize control rights as the most important factor in determining this issue. The distinction is naturally along a continuum. For instance, oDesk and Task Rabbit are almost pure platforms and therefore their professionals are clearly independent contractors: they set their own hourly rates, hours of work and choose which projects to work on. In contrast, other ﬁrms are positioned in the gray area in-between pure platforms (enabling independent contractors) and pure traditional ﬁrms (employing workers), because they control certain key decisions. For example, Uber and Lyft notoriously control the price of rides, although they allow drivers ﬂexibility in choosing the cars they drive (subject to some minimal requirements) and their work hours. Handy, which connects users with house cleaners, provides a more extreme example. Handy not only controls price, but also imposes numerous and detailed requirements on how cleaners should present themselves and complete the jobs (e.g. mandatory Handy insignia on clothing, how to communicate with customers before and after service, etc.). Unsurprisingly, Uber, Lyft and Handy are all currently involved in lawsuits aimed to determine whether their control over various aspects of the professional-customer transaction implies that they should treat professionals as employees instead of independent contractors.13 This would carry signiﬁcantly higher costs in the form of employee beneﬁts. One can interpret this issue in terms of our framework as implying that taking control over transferable actions in T -mode carries additional ﬁxed costs, which the ﬁrm does not have to incur in P -mode. This simply shifts the tradeoﬀ in favor of the P -mode, but all other considerations in our model continue to hold. Needless to say, there are other considerations that are relevant to the policy debate regarding the
See for example Justin Fox “Uber and the Not-Quite-Independent Contractor” Bloomberg, June 30, 2015. See Vinay Jain “Investors Must Confront The On-Demand Economy’s Legal Problem, Part 1” TechCrunch, January 12, 2015.
13 12

27

proper boundary between employees and independent contractors, but are not captured in our model: the impact of the work being done and the control rights associated with it on professionals’ outside options, the intensity of competition faced by the relevant ﬁrms/platforms, etc. Incorporating some of these aspects into the analysis provides a promising avenue for future research based on our work in this paper. There are several other directions in which our work can be extended. One would be to allow the ﬁrm to charge customers a ﬁxed access fee in both modes and study the impact of such a fee on the tradeoﬀ between the two modes. Such a fee is used by some ﬁrms selling some basic hardware or software to consumers, which is required in order to access the relevant services or products (e.g. video-game consoles). Another direction would be to study competition among ﬁrms that can each choose between the traditional mode and the platform mode, potentially leading to equilibria in which ﬁrms compete with diﬀerent models. Finally, one could pursue the analysis of hybrid modes along the lines brieﬂy described in Section 6 by incorporating cost or information asymmetries between the ﬁrm and the professionals and determining their eﬀect on the optimal allocation of control rights.

References
[1] Evans, D. S., A. Hagiu and R. Schmalensee (2006) Invisible Engines: How Software Platforms Drive Innovation and Transform Industries, Cambridge, MA: The MIT Press. [2] Gawer A. and M. Cusumano (2002) Platform Leadership: How Intel, Microsoft, and Cisco Drive Industry Innovation, Boston, MA: Harvard Business School Press. [3] Gawer, A. and R. Henderson (2007) “Platform Owner Entry and Innovation in Complementary Markets: Evidence from Intel,” Journal of Economics & Management Strategy, 16 (1), 1-34. [4] Gibbons, R. (2005) “Four Formal(izable) Theories of the Firm?” Journal of Economic Behavior & Organization, 58, 200–245. [5] Grossman, S. and O. Hart (1986) ”The Costs and Beneﬁts of Ownership: A Theory of Vertical and Lateral Ownership,” Journal of Political Economy, 94 (4), 691–719. [6] Hagiu, A. and J. Wright (2013) “Do You Really Want to Be an eBay?” Harvard Business Review, 91 (3), 102–108. [7] Hagiu, A. and J. Wright (2015a) “Marketplace or Reseller?” Management Science, 61 (1), 184– 203. [8] Hagiu, A. and J. Wright (2015b) “Multi-sided Platforms” International Journal of Industrial Organization, forthcoming. [9] Hart, O. and J. Moore (1990) ”Property rights and the nature of the ﬁrm,” Journal of Political Economy, 98, 1119–1158. [10] Holmstrom, B. (1982) ”Moral Hazard in Teams,” Bell Journal of Economics, 13 (2), 324-340. 28

[11] Holmstrom, B. and P. Milgrom (1987) ”Aggregation and Linearity in the Provision of Intertemporal Incentives,” Econometrica, 55 (2), 303-328. [12] Romano, R. E. (1994) “Double Moral Hazard and Resale Price Maintenance,” Rand Journal of Economics, 25 (3), 455–466. [13] Rysman, M. (2009) “The Economics of Two-Sided Markets,” Journal of Economic Perspectives, 23, 125-143.

Appendix
Proof of Proposition 1
The optimal contract Φ∗ (R) chosen by the ﬁrm in T -mode solves ΠT ∗ = max
Φ(.),a,e,E

R (a, e, E) − Φ (R (a, e, E)) − ca (a) − cE (E)

subject to  a   a = arg maxa {R (a , e, E) − Φ (R (a , e, E)) − c (a )}    e = arg max {Φ (R (a, e , E)) − ce (e )} e  E = arg maxE R (a, e, E ) − Φ (R (a, e, E )) − cE (E )     0 ≤ Φ (R (a, e, E)) − ce (e) . We start by proving the following lemma.

Lemma 1 Φ∗ (R) must be continuous and diﬀerentiable at R∗ = R (a∗ , e∗ , E ∗ ), where R∗ is the revenue that results from the optimization problem above. Proof. Suppose Φ∗ is discontinuous at R∗ and limR→R∗− Φ∗ (R) > limR→R∗+ Φ∗ (R). Then E ∗ = arg max R (a∗ , e∗ , E) − Φ∗ (R (a∗ , e∗ , E)) − cE (E)
E

implies Φ∗ (R∗ ) = limR→R∗+ Φ∗ (R), because otherwise Φ∗ (R∗ ) > limR→R∗+ Φ∗ (R), so the ﬁrm could proﬁtably deviate to E ∗ + ε, with ε suﬃciently small. But then we must have e∗ = 0, since otherwise e∗ > 0 and the professional could proﬁtably deviate to e∗ − ε with ε suﬃciently small. If e∗ = 0 then it must be that Φ∗ (R∗ ) = 0 and therefore (a∗ , E ∗ ) = arg max R (a, 0, E) − ca (a) − cE (E) .
a,E

(17)

In this case the ﬁrm could switch to the following linear contract: Φε (R) = εR + ce (e (ε)) − εR (a (ε) , e (ε) , E (ε)) ,

29

where ε > 0 is suﬃciently small and (a (ε) , e (ε) , E (ε)) is a solution to   a (ε) = arg maxa {R (a, e (ε) , E (ε)) − Φε (R (a, e (ε) , E (ε))) − ca (a)}  e (ε) = arg maxe {Φε (R (a (ε) , e, E (ε))) − ce (e)}   E (ε) = arg maxE R (a (ε) , e (ε) , E) − Φε (R (a (ε) , e (ε) , E)) − cE (E) . Denote the ﬁrm proﬁts that result from oﬀering contract Φε by ΠT (ε) ≡ R (a (ε) , e (ε) , E (ε)) − ca (a (ε)) − ce (e (ε)) − cE (E (ε)) . Clearly, (a (0) , e (0) , E (0)) = (a∗ , 0, E ∗ ) and ΠT (0) = ΠT ∗ . We can then use (17), the deﬁnition of e (ε) and assumption (a2) to obtain ΠT (0) = (Ra (a∗ , 0, E ∗ ) − ca (a∗ )) aε (0) + (Re (a∗ , 0, E ∗ ) − ce (0)) eε (0) ε a e + RE (a∗ , 0, E ∗ ) − cE (E ∗ ) Eε (0) E ∗ , 0, E ∗ ) Re (a = Re (a∗ , 0, E ∗ ) > 0, ce (0) ee where ce = ee
d2 ce . de2

Thus, if limR→R∗− Φ∗ (R) > limR→R∗+ Φ∗ (R) then the ﬁrm can proﬁtably deviate to Φε (R) for ε small enough, which contradicts the optimality of Φ∗ (R). The other possibility is limR→R∗+ Φ∗ (R) > limR→R∗− Φ∗ (R). Then e∗ = arg maxe {Φ∗ (R (a∗ , e, E ∗ )) − ce (e)} implies Φ∗ (R∗ ) = limR→R∗+ Φ∗ (R), because otherwise Φ∗ (R∗ ) < limR→R∗+ Φ∗ (R), so the professional could proﬁtably deviate to e∗ + ε, with ε suﬃciently small. But then we must have E ∗ = 0, otherwise the ﬁrm could proﬁtably deviate to E ∗ − ε with ε suﬃciently small. If a is not price then Ra (a, e, E) > 0 for all (a, e, E) so the same logic implies a∗ = 0. We must then have ΠT ∗ = R (0, e∗ , 0) − ce (e∗ ) = max {R (0, e, 0) − ce (e)} .
e

(18)

This cannot be optimal. Indeed, the ﬁrm could switch to the linear contract Φε (R) = (1 − ε) R + ce (e (ε)) − (1 − ε) R a (ε) , e (ε) , E (ε) , where ε > 0 is suﬃciently small and a (ε) , e (ε) , E (ε) is a solution to   a (ε) = arg maxa R a, e (ε) , E (ε) − Φε R a, e (ε) , E (ε) − ca (a)    e (ε) = arg maxe Φε R a (ε) , e, E (ε) − ce (e)     E (ε) = arg maxE R (a (ε) , e (ε) , E) − Φε (R (a (ε) , e (ε) , E)) − cE (E) .

30

Denote the ﬁrm proﬁts that result from oﬀering contract Φε by ΠT (ε) ≡ R a (ε) , e (ε) , E (ε) − ca (a (ε)) − ce (e (ε)) − cE E (ε) . Clearly, a (0) , e (0) , E (0) = (0, e∗ , 0) and ΠT (0) = ΠT ∗ . Using (18), the deﬁnitions of a (ε) and

E (ε) and assumption (a2), we obtain ΠT (0) = Ra (0, e∗ , 0) aε (0) + (Re (0, e∗ , 0) − ce (e∗ )) eε (0) ε e +RE (0, e∗ , 0) Eε (0) Ra (0, e∗ , 0) RE (0, e∗ , 0) > 0, = Ra (0, e∗ , 0) + RE (0, e∗ , 0) ca (0) cE (0) aa EE where ca = aa of Φ∗ (R). If a is price then ca = 0 so we must have ΠT ∗ = R (a∗ , e∗ , 0) − ce (e∗ ) ≤ max {R (a, e, 0) − ce (e)} .
a,e d2 ca da2

and cE = EE

d2 cE . dE 2

Thus, the ﬁrm can proﬁtably deviate to Φε (R) for ε small enough, which contradicts the optimality

Once again, it is straightforward to verify that the ﬁrm could proﬁtably deviate to Φε (R) for ε small enough. We have thus proven that limR→R∗+ Φ∗ (R) = limR→R∗− Φ∗ (R), so Φ∗ is continuous at R∗ . Suppose now that Φ∗ is non-diﬀerentiable at R∗ and limR→R∗+ Φ∗ (R) > limR→R∗− Φ∗ (R). This R R implies e∗ = 0, otherwise we must have 0 ≥ lim {Φ∗ (R (a∗ , e, E ∗ )) Re (a∗ , e, E ∗ ) − ce (e)} > lim {Φ∗ (R (a∗ , e, E ∗ )) Re (a∗ , e, E ∗ ) − ce (e)} , R e R e
e→e∗+ e→e∗−

so setting e slightly below e∗ would violate e∗ = arg maxe {Φ (R (a∗ , e, E ∗ )) − ce (e)}. If e∗ = 0 then we must have Φ∗ (R∗ ) = 0 (recall ce (0) = 0) and therefore (a∗ , E ∗ ) = arg max R (a, 0, E) − ca (a) − cE (E) .
a,E

But then we can apply the same reasoning as above to conclude that the ﬁrm could proﬁtably devaite to the linear contract Φε (R) for ε small enough. Suppose instead limR→R∗+ Φ∗ (R) < limR→R∗− Φ∗ (R). This implies E ∗ = 0, otherwise we must R R have 0 ≤ < lim RE (a∗ , e∗ , E) (1 − Φ∗ (R (a∗ , e∗ , E))) − cE (E) R E RE (a∗ , e∗ , E) (1 − Φ∗ (R (a∗ , e∗ , E))) − cE (E) , R E

E→E ∗− E→E ∗+

lim

31

so setting E slightly above E ∗ would violate E ∗ = arg maxE R (a∗ , e∗ , E) − Φ (R (a∗ , e∗ , E)) − cE (E) . If action a is not price then ca = 0 and Ra > 0, therefore the exact same reasoning applies to a∗ and leads to a∗ = 0. This would mean that ΠT ∗ = R (0, e∗ , 0) − ce (e∗ ) ≤ max {R (0, e, 0) − ce (e)} .
e

We have already proven above that this cannot be optimal. If action a is price then ca = 0 and ΠT ∗ = R (a∗ , e∗ , 0) − ce (e∗ ) ≤ max {R (a, e, 0) − ce (e)} .
a,e

In this case, we have proven above that the ﬁrm could do strictly better with the linear contract Φε (R) for ε small enough. We conclude that Φ∗ (.) must be continuous and diﬀerentiable at R∗ . The lemma implies that (a∗ , e∗ , E ∗ ) solve   Ra (a∗ , e∗ , E ∗ ) (1 − Φ∗ (R∗ )) = ca (a∗ ) a R  Re (a∗ , e∗ , E ∗ ) Φ∗ (R∗ ) = ce (e∗ ) e R   RE (a∗ , e∗ , E ∗ ) (1 − Φ∗ (R∗ )) = cE (E ∗ ) . R E Let then t∗ ≡ 1 − Φ∗ (R∗ ) and F ∗ ≡ (1 − t∗ ) R∗ − Φ∗ (R∗ ). Clearly, the linear contract Φ (R) = R (1 − t∗ ) R − F ∗ can generate the same stage 2 equilibrium as the initial contract Φ∗ (R). Furthermore, both Φ∗ (R) and Φ (R) cause the professional’s participation constraint to bind and therefore result in the same proﬁts for the ﬁrm. A similar proof applies to the case when the ﬁrm chooses the P -mode.

Proof of Proposition 5
Consider ﬁrst the T -mode. By (a6’) it is optimal for the ﬁrm to induce all N professionals to join, so the optimal contract Φ∗ (.) solves
N

ΠT ∗ =

max
Φ(.),E,(ai ,ei )i=1,..,N i=1

[R (ai , σ (a−i ) , ei , E) − Φ (R (ai , σ (a−i ) , ei , E)) − ca (ai )] − cE (E)

subject to
N i=1 R ai , σ a−i , ei , E − Φ R ai , σ a−i , ei , E ) ei = arg maxei {Φ (R (ai , σ (a−i ) , ei , E)) − ce (ei )} for all i ∈ {1, .., N }

(a1 , .., aN ) = arg max(a

1 ,..,aN

− ca (ai )

E = arg maxE 0≤

N i=1 [R (ai , σ (a−i ) , ei , E ) − Φ (R (ai , σ (a−i ) , ei , E Φ (R (ai , σ (a−i ) , ei , E)) − ce (ei ) for all i ∈ {1, .., N }

))] − cE (E )

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By assumption (a6’), we know that the solution to this program is symmetric, so we can write ΠT ∗ = max
Φ(.),E,a,e

− − N [R (a, σ (→N −1 ) , e, E) − Φ (R (a, σ (→N −1 ) , e, E)) − ca (a)] − cE (E) a a

(19)

subject to − − R (a , σ (→N −1 ) , e, E) − Φ (R (a , σ (→N −1 ) , e, E)) a a → − − + (N − 1) (R (a, σ (a , a N −2 ) , e, E) − Φ (R (a, σ (a , →N −2 ) , e, E))) − ca (a ) a → − e (e )} e = arg maxe {Φ (R (a, σ ( a N −1 ) , e , E)) − c − − E = arg max N [R (a, σ (→ a ) , e, E ) − Φ (R (a, σ (→ a ) , e, E ))] − cE (E ) a = arg maxa
E N −1 N −1

− 0 ≤ Φ (R (a, σ (→N −1 ) , e, E)) − ce (e) a Let then (a∗ , e∗ , E ∗ ) denote the symmetric outcome of this optimization problem. Also deﬁne R∗ ≡ → − R a∗ , σ a∗ N −1 , e∗ , E ∗ . Lemma 2 Φ∗ (.) is continuous and diﬀerentiable at R∗ . Proof. Due to symmetry, the proof is almost identical to the one of lemma 1, so we omit it. In particular, assumption (a5’) ensures that the deviation contracts Φε and Φε yield the desired result, just like in lemma 1. The lemma and program (19) imply that (a∗ , e∗ , E ∗ ) solve  − −  (1 − Φ∗ (R∗ )) Ra a∗ , σ →N −1 , e∗ , E ∗ + (N − 1) σa (a∗ ) Rs a∗ , σ →N −1 , e∗ , E ∗ a∗ a∗  R   → − Φ∗ (R∗ ) Re a∗ , σ a∗ N −1 , e∗ , E ∗ = ce (e∗ ) e R   −  ∗  N (1 − Φ∗ (R∗ )) RE a∗ , σ →N −1 , e∗ , E ∗ = cE (E ∗ ) a R E = ca (a∗ ) a

Let then t∗ ≡ 1 − Φ∗ (R∗ ) and F ∗ ≡ (1 − t∗ ) R∗ − Φ∗ (R∗ ). Clearly, the linear contract Φ (R) = R (1 − t∗ ) R − F ∗ can generate the same stage 2 symmetric Nash equilibrium (a∗ , e∗ , E ∗ ) as the initial contract Φ∗ (R). Furthermore, both Φ∗ (R) and Φ (R) cause the professionals’ participation constraint to bind and therefore result in the same proﬁts for the ﬁrm. A similar proof applies to the case when the ﬁrm chooses the P -mode.

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