Justin Taylor has written a short but helpful article, putting together the thoughts of other scholars on this theme:

Does your theology have categories by which to understand both God’s hardening of Pharaoh’s heart and then Pharaoh’s subsequent self-hardening? It’s a good test-case for biblically understanding divine sovereignty and human responsibility.

Here is a quick run-down of the key biblical data:

• Three times Yahweh declares that he will harden Pharaoh’s heart (Ex. 4:21; 7:3; 14:4).

• Six times Yahweh actually hardens Pharaoh’s heart (Ex. 9:12; 10:1; 10:20; 10:27; 11:10; 14:8).

• Seven times the hardening is expressed as a divine passive with Yahweh as the implied subject, i.e., Pharaoh’s heart “was hardened” by Yahweh (Ex. 7:13; 7:14; 7:22; 8:19; 9:7; 9:35; 14:5).

• And three times we are told that Pharaoh hardened his own heart (Ex. 8:15; 8:32; 9:34).
Divine-hardening and self-hardening are interwoven, but the God’s action is primary and initiatory: the first five citations (in Exodus 4 and 7) all focus on God’s action; the important point of Pharaoh’s self-hardening only appears in the three verses of Exodus 8 and 9.

The Apostle Paul famously reflected on the theological implications of this in Romans 9, using it to demonstrate the power of God’s mercy over the human will. Note the inclusio (or literary envelope) in Romans 9:16-18, including his quote of Exodus 9:16 on God’s purpose in hardening Pharaoh’s heart:
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