Enabling Versus Controlling - Online Appendix

Section 1 of this online appendix contains the proof of the technical Lemma (Lemma 2) used in the Proof of Lemma 1 in the main paper, which states that Ω* (.) is continuous and differentiable at R*. Section 2 provides the linear example with cost differences between the principal and the agent in choosing the transferable actions, based on the discussion in Section 4.4 of the main paper. We show that it can still be optimal to allocate all transferable decision rights to the same party even when the other party has a cost advantage for some of these decisions. Section 3 contains the proof of the claim in Section 5 of the main paper that if agents can cooperate in P -mode, they can overcome any attempt by the principal to use team payments to eliminate the double-sided moral hazard problem. Section 4 contains the proof that Lemma 1 still applies in the presence of spillovers. Section 5 provides the derivation of the closed form solutions for the linear model with spillovers used in Section 5.1 of the main paper. Section 6 does likewise for the model with pricing and spillovers used in Section 5.2 of the main paper. Section 7 establishes the result stated at the end of Section 5.1 in the main paper, namely that Proposition 5 continues to hold even if prices are endogenous and contractible, and there are production costs. Finally, Section 8 analyzes the hybrid case in a setting with multiple agents, in which some agents can operate in P -mode and others in A -mode.


Linear example
We directly conduct the calculations for the linear example with N > 1 agents and spillovers, i.e.
where the revenue generated by agent i is (1) The results for the case N = 1 are then simply derived by setting x = 0 and N = 1.
Consider first the E-mode.The payoff to agent i from working for the firm is which implies that the level of effort chosen by each agent in the second stage is e E (t) = γ (1 − t) .
In E-mode, the firm sets a 1 , ..., a N and I to maximize its second stage revenues (wages are paid in the first stage): implying the firm's optimal choices are a E (t) = θt The fixed fee T is set to render each agent indifferent between working for the firm and her outside option, so the expression of E-mode profits as a function of t is (2) Maximizing (2) with respect to t implies the optimal variable fee in E-mode is which is positive but smaller than 1.With this optimal variable fee, the resulting profits in E-mode are (3) Consider next the P -mode.The payoff to an individual agent joining the platform is Individual agents maximize their second stage payoff by choosing The firm's second stage profits in P -mode are which the firm maximizes over I, leading to Stepping back to the first stage, the firm sets T to equalize the agents' net payoff to their outside option.Total firm profit in P -mode as a function of t is then The optimal variable fee is Resulting profits in P -mode are Comparing ( 3) with ( 5), the P -mode is preferred if and only if If there are no spillovers, i.e. x = 0, then this condition simplifies to we apply the envelope theorem to expressions (2) and ( 4) and obtain Since 0 < t E * , t P * < 1 and t (2 − t) is increasing in t for t ∈ [0, 1], we conclude that 2 Linear example: endogenous price and production costs We now extend the linear example from the previous section with x = 0 and N = 1, by allowing the firm to also set a price in the contracting stage, along with the fees (t, T ), and by also adding a production cost.We will establish the result stated at the end of Section 4.3 in the main paper, i.e.
that Proposition 4 continues to hold in this case.
The revenue generated by the agent is now where d ≥ 0 is a constant marginal production cost, p is the price chosen by the firm and D 0 is some baseline level of demand.Fixed costs are still quadratic First, we show that whether the production cost is incurred by the firm or the agent does not affect profits in either mode.In E-mode, if the firm incurs the production cost, then the maximization problem is If instead the agent incurs the production cost, then the maximization problem is By making the change of variables t ≡ t − d p , the second maximization problem becomes the same as the first.
Similarly, in P -mode, if the firm incurs the production cost, then the maximization problem is If instead the agent incurs the production cost, then the maximization problem is Again, by making the change of variables t = t − d p , the second maximization problem becomes the same as the first.Thus, in our setting it is irrelevant which party actually incurs the production cost.
Solving the program above in E-mode, we obtain Holding p fixed and optimizing over t, we obtain Substituting this back into Π E * , the program becomes where Π E * is given by (3) with x = 0 and N = 1.Similarly, solving the program above in P -mode, we have Holding p fixed and optimizing over t, we obtain Substituting this back into Π P * , the program becomes where Π P * is given by ( 5) with x = 0 and N = 1.
Comparing the last expressions of Π P * and Π E * , we can conclude that so the introduction of p and d does not affect the trade-off determined in Proposition 4 in the main paper.
3 Private benefits (N = 1) Revenue is R (a, e, I).The firm derives private benefits Y (a), whereas the agent derives private benefits y (a).We can state an extended version of Proposition 3 in the main paper.
Proposition 1 Suppose R (a, e, I) is supermodular in its arguments, Y (a) = Y a and y (a) = ya, where Y and y are positive constants.Then: (i) if Y > y and where (a, e, I) is the unique solution to the three equations We first prove two preliminary lemmas.
Proof.Note that the solution (a, e, I) corresponds to a game in which there are three players, which seek to maximize respectively Since R is supermodular, it is easily seen that the game is supermodular with payoffs having strictly increasing differences in the actions (a, e, I) and the parameters (τ 1 , τ 2 , τ 3 , τ 4 ).From standard supermodularity results 1 , we know that an increase in any of the parameters (τ 1 , τ 2 , τ 3 , τ 4 ) will increase each of the solutions (a, e, I) in a weak sense.To obtain the strict comparative static result, note that the solution is defined by the set of equations defined in ( 6).This means that if τ i increases for some 1 See, for instance, X. Vives (1999), Oligopoly Pricing: Old Ideas and New Tools, Cambridge, Mass.: MIT Press.
i ∈ {1, 2, 3, 4} and a, e or I does not strictly increase, then neither a, e and I can change since none can decrease.But if (a, e, I) all remain unchanged, then, since τ i is higher, the first-order conditions (6) can no longer hold.Thus, at least one among (a, e, I) must strictly increase.
Proof.We have where we have used ( 6) to replace c a a (a), c e e (e) and c I I (I).By assumption, R a > 0, R e > 0 and R I > 0, and from Lemma 1, we know that da dτ 1 ≥ 0, de dτ 1 ≥ 0 and dI dτ 1 ≥ 0, with at least one strict inequality.Thus, we can conclude that dΠ dτ 1 > 0 when (τ 1 , τ 2 , τ 3 , τ 4 ) ∈ [0, 1) 4 .A very similar reasoning proves that dΠ dτ i > 0 for i = 2, 3, 4 when (τ 1 , τ 2 , τ 3 , τ 4 ) ∈ [0, 1) 4 .We can now use these lemmas to prove Proposition 1. Suppose y > Y and t E * ≤ 1/2.Then the firm could strictly improve on E-mode profits Π E * by giving up control over the transferable action a to the agent and keeping the variable fee unchanged, equal to t E * .To see this, note that the change in profits is If t E * > 0, then this difference is positive by Lemma 2, because 0 < t E * < 1/2 and y > Y imply Thus, in all cases we have Similarly, denote by t P * the optimal transaction fee in E-mode and suppose t P * > 1/2 and Y > y.
Then the firm could strictly improve on P -mode profits by taking control over the transferable action a and keeping the variable fee unchanged, equal to t P * .To see this, note that the change in profits is If t P * < 1, then this difference is positive by Lemma 2, because 1 > t which is positive due to assumptions (a1)-(a4).
Thus, in all cases we have The claim of Proposition 1 says that when the transferable action has a higher impact on the firm's private benefits relative to the agent's, the firm would never find it profitable to function in P -mode and charge variable fees above 50%.Similarly, when the transferable action has a higher impact on the agent's private benefits relative to the firm's, the firm would never find it profitable to function in E-mode and pay a bonus above 50%.
This can be re-stated in a more empirically useful way.To do so, define which is the optimal variable fee charged by the firm in the optimal mode.We obtain the following extension of Corollary 1 in the main paper.Thus, if the agent obtains more than 50% of variable revenues and the agent's private benefits are more important, then the firm should be functioning in P -mode, and not in E-mode.This suggests (as in Corollary 1 in the main paper), that only observing whether agents keep more or less than 50% of variable revenue can reveal the mode under which a firm is operating.However, the result requires that the party receiving more than 50% of variable revenue also has the more important private benefits.In Section 7 below we show numerically that even if this is not the case, knowing whether agents receive more or less than 50% of variable revenues still allows us to correctly predict the choice of mode most of the time.

Linear revenue function and linear private benefits
Again, we resort to the example with linear revenue function and quadratic cost functions: R (a, e, I) Private benefits are also assumed to be linear: where Y , y > 0.
In E-mode, the firm solves which leads to In P -mode, the firm solves Straightforward calculations yield which leads to Comparing the two profits, the P -mode is preferred to the E-mode if and only if

Linear revenue function and private benefits proportional to demand
The revenue and cost functions remain the same, but suppose now private benefits are proportional to the demand underlying the linear revenue function: where θa + γe + δI measures demand and the price is normalized to 1.
In E-mode, the firm solves which leads to In P -mode, the firm solves which leads to Comparing the two profits, the P -mode dominates the E-mode if and only if Thus, with this specification of private benefits, they have no impact on the trade-off between E-mode and P -mode.This result might seem surprising, particularly in the case when Y > 0 and y = 0: one would then expect the E-mode to always dominate because the firm takes the private benefit Y into account when setting a, whereas the agent does not.The reason this is not true is that in P -mode the firm needs to provide the agent with sufficient incentives for choosing both a and e, so it sets t P * sufficiently low (specifically, 1 − t P * > t E * + Y ) such that it compensates for the fact that the agent does not internalize the firm's private Y when setting a.

Cost asymmetries
Consider the linear example with one agent R (a, e, I) = θa + γe + δI and allow for different costs of the transferable action in the two modes: Straightforward calculations lead to Let us determine the effects of γ and δ on the tradeoff between the two modes.Using the envelope theorem, we obtain And we have Therefore, the magnitudes of moral hazard γ and δ have the expected effect on the trade-off.

Example with price as transferable decision and linear demand
Recall the revenue function and the assumptions made on parameters in the main paper: The fixed costs of agents' effort and firm's investment are quadratic: In E-mode, the payoff to agent i from working for the firm is which the agent optimizes over e i in the second stage (the fixed fee T is then taken as fixed).
The firm's payoff in the second stage is which the firm optimizes over p i and I.
Evaluating the corresponding first-order conditions at the symmetric equilibrium, we have Solving, we obtain Note that assumptions (7) ensure p E (t) > 0, e E (t) > 0 and I E (t) > 0.
The fixed fee T is just a transfer that renders each employee indifferent between working for the firm and their outside option, so the firm's profit is Plugging in the expressions of p E (t), e E (t) and I E (t) above, we obtain: In P -mode, an agent joining the platform chooses (p i , e i ) to maximize his second stage payoff while the firm chooses I to maximize its second stage revenues Evaluating the corresponding first-order conditions at the symmetric equilibrium, we have (−2θ + x) p P = d + γe P + δI P e P = (1 − t) γp P I P = tN δp P .
The fixed fee T renders each agent indifferent between joining the platform and his outside option, so the firm's profit in P -mode is for all permissible x, so we already know that 6 Hybrid mode across agents Recall the expression of firm profits when using a hybrid mode as a function of t E , t P , n : Optimizing over the three variables t E , t P , n yields the following first-order conditions (assuming interior solution in all three variables): Solving the first two first-order conditions above for t E , t P as functions of n, we obtain: This implies: • For each value of γ 2 in the set defined in the previous step, consider 100 equally spaced values of y from 0 to γ 2 and 100 equally spaced values of Y from 0 to δ 2 .This range of y and Y ensures that t E * , t P * ∈ [0, 1] 2 .
• Repeat the previous two steps for a range of different δ 2 as reported in Table 1 below.For each value of δ 2 , record the average values of t E * and t P * across all 990, 000 observations of γ 2 , y, Y defined in the previous steps and the fraction of these observations for which the prediction of Corollary 1 is true.The higher the percentage of observations for which the prediction of Corollary 1 is true, the more confident we can be in using the observed commissions received by agents (or variable fees charged to agents) to infer which mode the firm operates in.Specifically, if we observe agents keeping more than 50% of variable revenues, then the firm is most likely to be using P -mode; if we observe agents keep less than 50% of variable revenues, then the firm is most likely using the E-mode.The values in Table 1 make it clear that this inference is remarkably accurate for low values of δ 2 .

Spillovers
With spillovers rather than private benefits, we have the additional parameter x but no longer have the parameters y and Y .Also the parameter δ 2 becomes N δ 2 .The procedure is similar to the case with private benefits.
We normalize θ = 1 throughout.We proceed in two steps: • For a given value of N δ 2 , build the set of values of γ 2 such that t E * varies from 0.01 to 0.99 in 0.01 intervals (recall that t E * does not depend on x), and the set of values of x from −0.99 to 0.33 in 0.01 intervals (this is the range of spillovers such that the cross-effect of the transferable action is less than half as important in magnitude as the own effect of the transferable action). 3 3 Given that the own effect is θ − x, the cross-effect is x and θ = 1, this condition requires x < 1 2 (1 − x) for x > 0 and |x| < 1 2 (1 + |x|) for x < 0, which is equivalent to −1 < x < 1 3 .
• Repeat the previous step for a range of different N δ 2 as reported in Table 2 below.For each value of N δ 2 , record the average values of t E * and t P * across all 13, 167 observations of γ 2 , x defined in the previous step and the fraction of these observations for which the prediction of Corollary 1 is true.

τ 3
R e (a, e, I) = c e e (e) τ 4 R I (a, e, I) = c I I (I) .

Corollary 1
Suppose R (a, e, I) is supermodular in its arguments, Y (a) = Y a and y (a) = ya, where Y and y are positive constants.If y > Y and t * ≤ 1/2, then the P -mode is optimal.If Y > y and t * ≥ 1/2, then the E-mode is optimal.