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

  • Nina Baranchuk EMAIL logo , Seethu Seetharaman and Andrei Strijnev

Abstract

For many years, the movie industry has been characterized by a unique (compared to other industries) type of vertical contracting practice, called sliding-scale contracting whereby the distributor (studio) takes a much larger (usually around 70%) share of box-office revenues than the exhibitor (theater) in the week of a movie’s release, with the exhibitor’s share increasing, in gradual steps, over subsequent weeks. In this paper, we propose a game-theoretic model that provides a new rationale for these contracting choices. Specifically, we show that these contracts effectively resolve conflicts of interest between studios and theaters over movie release timing and display length, in a way that is beneficial for both parties. Our model also stipulates conditions under which sliding scale become dominated by aggregate deals, i.e. deals based on total rather than weekly box office revenue. The testable predictions based on these conditions can be used by future empirical research once the available evidence on the use of aggregate deals in practice goes beyond anecdotal.

Appendix

Lemma 2

Sharing Rules and Closing Timing

In equilibrium, closing times that correspond to opening timestomaximize the difference between the total box office and the total display costs generated by movie j,

tjcto=argmaxtctjotjc1+rDjttjo+1to,tcCjtargmaxtctjotjcΠjtto,tc.

Moreover, if equilibrium exists, then there is an equilibrium where the sharing rules are renegotiation-proof on the equilibrium path.

Proof

Without loss of generality, we can restrict our attention to equilibria with no renegotiation on the equilibrium path (there still may be renegotiation off equilibrium, for example, if one of the studios deviates from its equilibrium release time). To see that this is without loss of generality, suppose that, on the contrary, there were renegotiation on the equilibrium path from contracts θ to contracts θ. Then, there is an equivalent equilibrium where studios offer contracts θ at the outset and there is no further renegotiation. Note that the possibility of renegotiation still matters as it imposes a constraint on the initial contract to be renegotiation proof.

Consider an equilibrium with contracts θ, opening times to and closing times tc that has no renegotiation on the equilibrium path. Suppose that, contrary to the lemma’s statement, the equilibrium closing time tjc does not maximize tjotjcΠjtto,tc. Instead, the expression is maximized by tjc. This in particular implies that tjctjcΠjtto,tc>0. Note that we do not make any assumption on whether tjc is larger or smaller than tjc. When it is smaller, tjc<tjc, the above sum has a larger value specified as the lower boundary and a smaller value specified as the upper boundary. In our context it is sensible to interpret this notation as follows: when tjc<tjc, let tjctjcΠjtto,tctjctjcΠjtto,tc. Then, studio j can benefit by offering the theater a new contract with θjttjo+1=1+rCjt/Djt and αjttjo+1=0 for t=tjc,,tjc (which is an empty set if tjc<tjc), and θjttjo+1=1 for t>tjc. With this contract, the theater will switch to the closing time tjc thereby increasing the studio’s profit by tjctjcΠjtto,tc.           □

Proof of Lemma 1

Consider an equilibrium where the equilibrium renegotiation-proof contract has αjτ>0 for some τ2. 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,θ3j 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 θjαjτ 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

(31)αj1+θj1Dˉj11+τ2θjτDˉjτ1>αj1+θj1Dˉj12+τ2θjτDˉjτ2.
  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

θj1Dˉj11>θj1Dˉj12+maxθj1sum+Dˉj12Dˉj11

Using the definition of sum+Dˉj12Dˉj11, we can obtain that the above inequality implies that

θj1Dˉj11>θj1Dˉj12+τ2θjτDˉjτ2Dˉjτ1
=θj1Dˉj12+τ2θjτDˉjτ2τ2θjτDˉjτ1

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

θj1Dˉj12Dˉj11<maxθj1sum+Dˉj11Dˉj12

Using the definition of sum+Dˉj11Dˉj12, we can obtain that the above inequality implies that

θj1Dˉj12Dˉj11<τ2θjτDˉjτ1Dˉjτ2

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

Dˉj1d3jtqjqj+kt+d0tqjqj+q3jI3jt+kt,

where I3jt 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 d3jt who have seen the competing movie, and is also smaller when I3jt equals 1. Simultaneous release time t3jo=tjo for the competing movie 3-j results in the smallest possible value of zero for d3jt and sets I3jt 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.

d0maxtcto,tc=kq1+kT1T2kq1+q2+kT2
d0maxtcto,tc=kq1+kt2o1kq1+q2+kT1t2o+1kq2+kT2+t2oT11
d0maxtcto,tc=kq1+k1kq1+q2+kT1+1kq2+kT2T11kq1+kt2okq1+q2+kt2okq2+kt2o

The above is decreasing with t2o because

kq1+kq1+q2+kq2+k<1

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:

Acjtγjcjt2+τ=t+1tjcδτAEtcjτγjEtcjτ2,

where Et denotes expectation formed at the end of period t. According to eq. (16), we can use the equality cjt=SjtYjt+Yjt1 to rewrite the above as

ASjtYjt+Yjt1γjSjtYjt+Yjt12+AEtSjt+1Yjt+1+YjtγjEtSjt+1Yjt+1+Yjt2+τ=t+1tjcδτAEtSjτYjτ+Yjτ1γjEtSjτYjτ+Yjτ12

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

A+2γjSjtYjt+Yjt1+A2γjEtSjt+1Yjt+1+Yjt=0.

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

cjt=Etcjt+1.

This in particular implies that the period 0 expectation of future consumption is the same in all periods. Let cˉjEcjt 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:

τ=1δτcˉj =τ=tjotjcδτESjt

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

(32)t=1δtA11δτ=tjotjcδτESjtγj11δτ=tjotjcδτESjt2γj1δ2δ2tVSjt,

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

t=1δtA11δτ=tjotjcδτETjtγT11δτ=tjotjcδτETjt2γTγT1δ2δ2tVTjt
(33)=Td+τ=tjotjcδτCjt,

where Tjt=αjt+1θjtDjt. 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/1δ as

(34)AγjδtESjt1δγj2δtθjt1δVDjtEDjtλAγTδtETt1δγT2δtθt1δVDjtEDjt=0

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

λ=AγjδtESt1δγj2δt+1θt+11δVDt+1EDt+1AγTδtETt1δγT2δt+1θt+11δVDt+1EDt+1

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:

τ=tjotjcδτETjt=τ=tjotjcδταjt+1θjtEDjt>τ=tjotjcδτCjt.

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+1θjtEDjtCjt for ttjo,tjc and αjt+1θjtEDjt<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.            □

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Published Online: 2019-12-11
Published in Print: 2019-11-18

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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