“AMERICAN IDOL” judge RANDY JACKSON may be worth an estimated $40 million, but he’s a cheapskate dad, according to court records exclusively obtained by The ENQUIRER.

While Randy’s fortune has soared in recent years, his ex-wife Elizabeth had to fight to get boosts in child support that reflect his massive income, say sources.

When the two divorced in 1993, Randy agreed to pay $1,950 a month in child support based on his $250,000 annual income.

But in 2004, Elizabeth took Randy back to court and de­manded more money. She claimed he was making a whop­ping $1 million a year as an “American Idol” judge while she was scraping by on approxi­mately $30,000 a year.

Elizabeth told the court that she and her then-14-year-old daughter Taylor were living in a rundown, 2-bedroom rent­ed house in Studio City, Calif., while Randy was residing with his new wife Erika, now 42, and their two kids in a multimillion-dollar Encino mansion.

According to papers filed in court, Elizabeth’s rented house was uncomfortably cold at night, with paint peeling in the bathroom and an unreliable water heater.

But their home wasn’t the only problem, said a source who characterized Randy, 55, as “a bit of a tightwad.”

The source told The ENQUIRER : “If Liz needed additional fi­nancial help for their daughter, like tutoring or a hair appoint­ment, she had to write to his ac­countants to be reimbursed. She also asked Randy to reimburse her for Taylor’s lunch money – just $71 a month – and he even stalled on that.

“He eventually paid up, but his accountants were slow to respond and Liz needed to follow up with several faxes and phone calls.”

Elizabeth, 51, also complained to the court about not having enough money to pay for Taylor going out with friends to places like theme parks and concerts.

Court documents show Randy finally agreed to give Elizabeth $3,750 a month in child support for Taylor until her 18th birthday as well as cover the costs of her private schooling.

He further agreed to pay $100,000 toward her college education.

“It’s a start,” said the source. “But it’s still an insignificant amount in comparison to what he’s worth.”