Modulate research effort in inbound alliance

We address the R&D supply chain where a customer wants to sign an R&D contract with a supplier in an inbound open innovation style. Supplier has to decide how much effort put on contract: effort is assumed as nonobservable then it cannot be constrained contractually. We model the R&D process like a staged one, which allows for both an early contract and a late one. The overall committed effort depends on the alliance timing: earlier contracts will ask for a greater cumulated effort although they offer risk-sharing opportunity. On the other hand, later contracts see supplier with a stronger ex-ante bargaining power position, due to shorter (and then less risky) contract length and experience acquired in the previous period in her own. The third element (after timing and supplier effort) of the contract we account for is the royalty amount that can be prefigured only by the supplier being her effort unobservable. We model the problem assuming that when both partners exert the maximum affordable effort they are able to achieve the maximum innovation value (in this way we separate direct and indirect risk, and we assume no relational risk); moreover we assume a research production function convex for low effort and concave for high effort. These hypotheses make the optimal time to agree depending on the convexity of the production function: in case of convexity (low effort) it is more convenient for the supplier a late contract agreement for which the best response is an effort lower than in case of early contract and this means a lower innovation value. The opposite holds for high effort.