The budget income of the fund is a government fund with a specific purpose. Government fund is a kind of non-tax revenue with special purpose, which is levied on a specific group in order to achieve the policy objectives in a specific economic and social field. According to the regulations, when using funds, departments and units should strictly follow the channels of funds and use them within the prescribed expenditure scope and expenditure standards, and may not change the purpose of expenditure without authorization.

However, local government funds have different backgrounds and realistic conditions, which are the product of the anomie of China's financial system to a considerable extent and have been abused at one time or all the time. In practice, it is common to use fund income to pay debts locally.

The money used by local governments to repay debts should come from the public budget. Because public budget revenue includes debt revenue, such as government borrowing from foreign governments or international organizations, treasury bills and so on. According to the principle of consistent revenue and expenditure channels, the income of this department should also be used to pay off debts.

PS: The first of the two formulas you summarized is correct.