The generation of construction and demolition waste (CDW) is a problem for societies aspiring to sustainability. In this regard, governments have the responsibility to support the CDW recycling through subsidies. However, the information asymmetry, as well as the “dynamic nature” of the CDW recycling market, results in a number of barriers for the government to promote CDW recycling. In this paper, we establish a mathematical model that includes the government and the recycling enterprise in the presence of dual information asymmetry including the unknown recycling technology level and unobservable recycling efforts in one-stage and two-stage cooperation. Using the incentive theory, the static and dynamic optimal recycling incentive contracts of the government were designed, and the optimal decisions of the recycler were identified. A numerical simulation revealed that by designing reasonable contracts, the government can encourage the recycler to report the true technical level and achieve information screening. Furthermore, the subsidy of the high-tech recycler remained unchanged under different circumstances. However, the subsidy of the low-tech recycler was closely related to the probability of misreporting and the proportion of technology types. This finding suggests that the government and recycler are inclined towards long-term dynamic cooperation.