State Sen. Click Bishop remembers his first paycheck as a teenager in Fairbanks in the early 1970s. His boss explained the $10 deduction for the state’s so-called school head tax.

“That pays for your education,” the boss told his young employee.

“I’ve never forgotten that,” said Bishop.

The Legislature in 1980 abolished the small education tax, along with Alaska’s personal income tax and a tax on business gross receipts. The state was getting rich from oil and a majority of lawmakers saw little need for taxes.

Bishop, now in his 11th year in the Senate, is making his fourth a...