AppendixIn this appendix we show that the optimal degree of profit sharing in the backward contract is on the interval (0,1]. We do this in two parts. We first show the optimal contractual share to be non-negative. Assume the share is not positive. From expression (14), a* < 0 implies:Rearranging terms, we can express this result as:where the right-hand equality holds by the downstream first-order condition (13).The non-positive profit-share contradicts the optimal price set in the upstage industry, however, as c > PU by expression (12). Therefore, nonpositive market-share contracts contradict, whence the market-share of an upstream producer is positive.We next show the optimal contractual share to be consistent with values less than one. 7.