In NY even equal parenting time (the highest form of joint custody) does not automatically translate into no child support.
The law in NY is to "presume" that the parent with the greater gross income is the non-custodial parent and then apply the traditional formula of support based upon how many unemancipated children there are. Then the court may deviate based upon the shared and equal parenting time or any other relevant factor. Such factors may include the disparity in income between the parties, what the percentage allocation will be of child care, health insurance, co-pays, extra-curricular activities and any other relevant factors.
Of course, parents can mutually agree to waive the payment or receipt of child support because their incomes are relatively equal, or one parent has access to far greater assets than the other or because of the equitable distribution arrangement.
However, when there is no waiver and the incomes are quite different the court will likely fashion some or even the full amount of child support so that the children do not have to live in one household with minimal income and in luxury in the other household.
For example one parent earns $90,000 per year and the other parent earns $100,000 per year then the payment of support if there are two children could create a windfall for the lower income parent. By contrast, if one parent earns $90,000 and the other parent earns $700,000 then the court will likely require substantial child support and may also require support over the "statutory cap" which is currently base upon combined parental income of $193,000
The calculation of child support even with a "formula" is not always automatic and there are calculation nuances and arguments to request the court to use its discretion up or down. In such situations, experienced matrimonial counsel is critical to obtaining a good result. Experienced family law counsel will have knowledge of caselaw and how courts have handled similar income differences previously.