The production tax credit (PTC) promotes wind energy development, reduces power generation costs, and can affect merchants' joint economic dispatch, particularly for electricity merchants with both energy storage and wind farms. Two common PTC policies are studied - in the first policy, a wind farm receives PTC by selling wind generation to the market and its storage can be used to store energy from the wind generation and energy purchased from the grid but the energy released from the storage cannot receive PTC; in the second policy, the energy released from the storage can also qualify for PTC but purchasing energy from the grid is not allowed. We then employ dynamic programming to study merchants' optimal decision-making while considering PTC and the physical characteristics of storage systems. We analytically show that the state of charge (SOC) range can be segmented into different regions by SOC reference points under two PTC policies. The merchant's optimal action can be conveniently and uniquely determined based on the region within which the current SOC falls. Moreover, this study illustrates that PTC could substantially alter the optimal scheduling policy structures by affecting reference points and their relationships. The results showed that the frequencies for charging and discharging storage decisions decreased with an increase in PTC subsidy. Last, we confirm that, although the first policy allows merchants to buy electricity from the market, the second policy can bring more profits when the PTC is large at the current PTC rates. The findings can provide multistage decision-making guidance to electricity merchants in the wholesale power market.