Revenue Sharing Vertical Contracts in the Movie Industry: A Theoretical Analysis

Proof of Lemma 1

Consider an equilibrium where the equilibrium renegotiation-proof contract has αjτ∗>0 for some τ≥2. Then, there exists an equivalent equilibrium where studio j offers an alternative contract θ˜j that coincides with θj∗ everywhere except αjτ′=0, αj1′=αj1∗+αjτ; all remaining strategies are the same in the new and the original equilibria. The new contract θ˜j will leave all players with identical payoffs if they choose the same opening and closing strategies. According to Lemma 2, closing times following any contract and release strategies maximize the channel profit an d therefore will not change. Given these closing time strategies, the original equilibrium release time tjo∗ will bring studio j the same payoff under the new contract θ˜j as it did under θj∗. If there is an alternative release time tjo′ that results in a higher payoff under the new contract θ˜j, it means that, in the original equilibrium, studio j could have benefited by deviated to θ˜j, which contradicts the definition of the equilibrium. It remains to show that contracts θ˜j,θ3−j∗ are renegotiation-proof. With the new contract, a renegotiation proposal θj′ will be evaluated differently only if it induces the theater to close movie j earlier, prior to period τ. In this case, however, renegotiation to θj′ from contract θ˜j brings the same payoff to both studio j and the theater as renegotiation to θj′−αjτ from contract θj∗. Renegotiation-proofness of θj∗ then implies renegotiation-proofness of θ˜j.         □

Proof of Proposition 1

All sharing rules considered in this proof have fixed pay that is non-zero only in period 1 (as required in the proposition statement). The proof of all parts follows from comparing the studio’s profits from both release dates. Studio j prefers tjo1 if

1. Inequality eq. (31) is equivalent to eq. (21) if θj is an aggregate deal. Thus, when eq. (21) holds and θj is an aggregate deal, studio j prefers tjo1.

2. Suppose the demand satisfies eq. (22) and consider θj that satisfies eq. (24). From eq. (22), both the numerator and the denominator in eq. (24) are positive, and thus we can equivalently rewrite eq. (24) as

Using the definition of sum+Dˉj−12−Dˉj−11, we can obtain that the above inequality implies that

Rearranged, the above is equivalent to eq. (31). Thus, when eq. (22) holds and θj satisfies eq. (24), studio j prefers tjo1.

1. Suppose the demand satisfies eq. (23) and consider θj that satisfies eq. (25). From eq. (23), both the numerator and the denominator in eq. (8) are positive, and thus we can equivalently rewrite eq. (25) as

Using the definition of sum+Dˉj−11−Dˉj−12, we can obtain that the above inequality implies that

Rearranged, the above is equivalent to eq. (31). Thus, when eq. (23) holds and θj satisfies eq. (25), studio j prefers tjo1.                                                                                        □

Proof of Proposition 2

The release period expected demand for movie j is given by

where I3−jt is an indicator that equals 1 if movie 3–j is displayed in period t and equals zero otherwise. The above decreases with the number of consumers d3−jt who have seen the competing movie, and is also smaller when I3−jt equals 1. Simultaneous release time t3−jo=tjo for the competing movie 3-j results in the smallest possible value of zero for d3−jt and sets I3−jt equal to 1, and thus minimizes the above expression.

We next prove the second claim of the proposition. Keeping display run of both movies the same, we can evaluate the size of the population that will have seen no movie by the time both movies are closed as follows.

The above is decreasing with t2o because

Thus, the channel revenue increases when movie 2’s release date is moved further from that of movie 1 if closing times are chosen to keep display run constant. Constant display run means that direct display costs also do not change, and thus increasing revenue is equivalent to increasing profit. Optimal choice of closing times will further increase the channel profit.                                  □

Proof of Proposition 3

To ease exposition of this proof, in what follows, we suppress some notation whenever it does not cause confusion. We will start the proof by showing that, given a contract, a release time, and a closing time, which fully determine the studio and the theater’s (random) income streams, both the studio and the theater will optimally use savings to fully smooth their consumption. Knowing the optimal consumption policy for a given income stream will allow us to express the studio’s and the theater’s utilities as functions of these income streams. Taking release and closing times as given, we will then show that an aggregate deal maximizes the studio’s expected utility subject to the theater’s participation constraint. Finally, we will show that, for any given closing time, fixed payments in the aggregate contract can be chosen so that the contract is incentive-compatible with this closing time.

At the end of period t, after learning its income Sjt, studio j chooses savings Yjt to maximize the remaining expected utility, which, according to eq. (28), can be evaluated as follows:

where Et denotes expectation formed at the end of period t. According to eq. (16), we can use the equality cjt=Sjt−Yjt+Yjt−1 to rewrite the above as

In the above, only the first two terms depends on Yjt. Thus, the first order condition to the above with respect to Yjt produces

Again using eq. (29) for cjt, the above can be restated as

This in particular implies that the period 0 expectation of future consumption is the same in all periods. Let cˉj≡Ecjt denote this period 0 expectation resulting from the above optimal savings choices. We can evaluate cˉj by observing that the studio’s consumption is bound by its income. This period 0 budget constraint implies that the present value of all future expected consumption is equal to the present value of all expected future income:

from the above, we obtain that cˉj=11−δ∑τ=tjotjcδτESjt. Using this expression, we can write period 0 expected utility eq. (28) of studio j as

where Sjt=αjt+θjtDjt. Studio j will choose θjt to maximize the above subject to the theater’s participation constraint. Similar derivations of the optimal savings pattern for the theater imply that the theater’s participation constraint can be written as

where Tjt=−αjt+1−θjtDjt. Letting be the Lagrange multiplier, we can express the first order condition to the studio’s problem with respect to θjt (after dividing it by δtEDjt/1−δ as

From the similar first order condition with respect to θjt+1, we can express as

Substituting the above expression forλ into eq. (34), we obtain that θt=θt+1, that is the contract that achieves optimal risk-sharing given opening and closing times is an aggregate deal.

It remains to show that the aggregate deal derived above can be incentive-compatible with the assumed closing time. Observe that the fixed part of the contract, αjt, enters into the studio’s objective eq. (25) and the theater’s participation constraint eq. (32) as a part of a linear sum, ∑τ=tjotjcδταjt. Further, the theater’s participation constraint implies that the theater’s overall expected income from the movie strictly exceeds the theater’s display costs:

Therefore, without altering either the constraint eq. (35) or the value of the objective function eq. (32), studio j can set αjt to satisfy −αjt+1−θjtEDjt≥Cjt for t∈tjo,tjc and −αjt+1−θjtEDjt<Cjt for all other periods t. Such fixed payments used in the aggregate deal derived above result in a contract that is incentive-compatible with the closing time tjc.            □