DAN ZIELINSKI, director of the planetarium at Jenks High School in Oklahoma, whizzes through his greatest hits. First he projects onto its dome a 3D image of a human heart; next comes the Sistine Chapel, then the solar system. The planetarium is an impressive asset for a high school, as is its aquatic centre, with an Olympic-size pool and grandstand seating. But there is a hitch, says Bonnie Rogers of Jenks Public Schools: filling the new buildings with teachers is much harder than erecting them.

At once bountiful and hard-pressed, the school district covers a well-heeled suburb of Tulsa and spans the Arkansas river to take in part of the city proper. The financial paradox has a simple explanation. The shiny facilities were paid for by municipal bonds, but teachers are financed by the state, and similar top-ups for their salaries are not allowed. In Oklahoma, state education funding has withered: since the crash of 2008 spending per pupil has been slashed by 27% in real terms, the biggest fall in America. Some districts run only four days of classes a week. Teachers earn much more across the border in Texas. The area is an extreme case of a wider trend, in which cities and their residents find ways to cope with miserly state governments.

In Oklahoma, the squeeze is extreme. Cuts to income taxes, generous incentives for fracking companies and low oil prices have choked revenues. Overall, this year’s state budget is 15% lower than that of 2009; Medicaid and welfare have been pinched along with schools, as have state troopers, whose mileage is now circumscribed. Mary Fallin, the governor, acknowledges that something has to give. She wants to expand the tax base and raise rates on fuel and cigarettes. But the Republican-controlled legislature has yet to agree on a fix. The state’s voters are not helping: in a referendum last year they rejected a plan to add a percentage point to the sales tax to boost education spending. Gene Perry of the Oklahoma Policy Institute notes that rugged ideology is not the only obstacle. The state constitution specifies that revenue measures must be approved by 75% of both legislative chambers—requiring some bipartisan agreement—or by the people.

As G.T. Bynum, Tulsa’s new, babyfaced mayor, laments, his scope for manoeuvre is just as narrow. Oklahoma sets tight limits on cities’ use of property taxes, leaving them reliant on sales taxes. That is a volatile source of revenue—the dependence can lead, for instance, to police officers being laid off during a recession—and one now undermined by online shopping. Mr Bynum’s cousin and grandfather were also mayors of Tulsa, as, in the frontier years of 1899-90, was a great-great-grandfather: the six-shooter he carried is in a cabinet in Mr Bynum’s office. Not long afterwards Tulsa became the “oil capital of the world”, memorialising its glory in skyscrapers. Mr Bynum wants to revive the clout lost in the oil and telecoms busts.

Fiscal constraints make that tricky. Still, the state legislature is considering a change that would let cities raise property taxes to help pay for policing. Meanwhile Tulsa is making the best of a tough predicament through bond issues (like those in Jenks and other cities), as well as a sales-tax increase which, unlike the statewide proposal, was approved by local residents last year, when Mr Bynum was still a city councilman. As with other successful local referendums, it helped that the initiative came with concrete details about where the extra cash would go: on public transport, the police, and investments in museums and other public facilities. By the time of the vote, smiles Mr Bynum, “everyone was sick of hearing about it”.

The miniature culture wars fought between cities and states—such as North Carolina’s tussle with Charlotte over its anti-discrimination rules—are well known. The financial tensions between them are quieter but as important. “Money is usually the main problem,” says Larry Jones of the United States Conference of Mayors, and especially divisive in lean times.

In this stand-off Tulsa, like other American former boomtowns, benefits from the afterglow of industrial wealth. Several times its tycoons have ridden to its rescue: to supply its water, to build a bridge to connect it with oilfields, and to buy the land for its airport. These days country-music stars live in some of the oil barons’ grand villas but, by way of compensation for the economic pendulum, the paternalism lives on. “Philanthropy is an industry here in Tulsa,” says Mr Bynum.

The George Kaiser Family Foundation, for example, has renovated several blocks in the city centre, and set up a diversion scheme for female prisoners and an early-years education programme. It has contributed $200m for a new 100-acre park on the river, while raising the same amount from other donors. Riding around the muddy construction site, Jeff Stava, the project’s boss, points to where the splash zone, skate park and giant adventure playground will be, and the stretch of water where perching pelicans will soon be ousted by rafts and kayaks. After the huge growth of the 20th century, but a tentative start to this one—the population is stagnating at around 400,000—the aim is to make Tulsa a place professionals will move to. The city is stumping up the money for a footbridge near the park.