The problem of time-consistency of cooperative solutions is investigated in the paper. This problem was stated by Petrosyan L.A. in 1977 for differential games with finite time horizon. In the paper the modification of the game with finite time horizon is considered in the sense that the game has random time horizon. The Shapley value is used as an optimality principle under cooperative behavior of the players. For this formulation the definition of the imputation distribution procedure (IDP) is given and the analytic formula for IDP is derived. Moreover in the paper the irrational behavior proofness condition by D.W.K. Yeung (2006) is modified for problem with random duration. The tool is based on using IDP. Theoretical results are illustrated by an example of differential game of non-renewable resource extraction.