For tax purposes, there is no such thing as joint custody, regardless of what your legal agreement says. The requirement, to be custodial parent, is that the child lived with you MORE than 50% of the time. One of you has to be the custodial parent and the other the non-custodial parent. The IRS expects you to count the nights the child sleeps at each home.
In the rare case (could probably only happen in a leap year like 2016), where the time that each parent has the child is exactly equal, then the parent with the higher AGI is the custodial parent, for the purpose of determining who has first priority on claiming the child as a dependent. But then neither parent can claim a Qualifying Child dependent, for some tax benefits because neither parent had the child the required MORE than half the year. (no earned income credit, based on that child, and the child would not qualify the parent for Head of Household filing status).
That said, essentially you can decide between you who will claim the child. And, if the number of nights is equal, you can decide between you who is the custodial parent. As long as you can agree, the IRS most likely won't question it. ideally, you arrange you custody schedule so that the parent claiming to be the custodial parent actually has him more than half the nights.
The custodial parent has first priority on claiming the children on his/her taxes; regardless of the amount of support provided by the non-custodial parent. The non-custodial parent can only claim the child as a dependent if the custodial parent gives permission (on form 8332) or if it's spelled out in a pre 2009 divorce decree. The requirement, to be custodial parent, is that the child live with you MORE than 50% of the time, regardless of what your legal agreement says*.
There is a way to split the tax benefits. For future negotiations with the other parent (and maybe even for this year) the following info may be of use:
There is a special rule in the case of divorced & separated (including never married) parents. When the non-custodial parent is claiming the child as a dependent/exemption/child tax credit; the custodial parent is still allowed to claim the same child for Earned Income Credit, Head of Household filing status, and day care credit. This "splitting of the child" is not available to parents who lived together at any time during the last 6 months of the year; then only one of you can claim the child for any tax reasons. The tax benefits may not be split in any other manner.
Note in particular that the non-custodial parent can never claim the Earned Income Credit, Head of Household filing status or the day care credit, based on that child, even when the custodial parent has released the exemption to him.
*If the child lived with each parent for an equal number of nights during the year, the custodial parent is the parent with the higher adjusted gross income (AGI) (2012 Pub 17 pg 28).