B IRMINGHAM, ENGLAND , is shaped like a kidney, taller than it is wide, curving eastwards slightly on its longer sides. It is contiguous, blob-like, sensible. Birmingham, Alabama, founded in 1871 and named after its English ancestor, looks as if it was imagined by a deranged computer, straight lines and sharp angles and missing bits in the middle (see illustration above). One pseudopodium extends to the west, long and thin, until it widens out a bit and ends in a box. To the east is a tumorous outgrowth, thin, then wide, then thin again, doubling back on itself several times.

From San Jose in the west to Savannah in the east, and from tiny Minot in North Dakota to sprawling Fort Worth in Texas, odd city maps can be found all across America. With some exceptions, these boundaries are administrative confections that make few allowances for geography, population density or common sense.

Los Angeles is a striking example. The county is home to 88 incorporated cities, ranging in population from 76 (Vernon) to 3.9m (Los Angeles itself). Beverly Hills (population 34,506) is its own city, with its own police force, fire department and school district. So is Santa Monica (92,495). Even Vernon has a police department, with some 50-odd staff. All told, the county has about 30 fire departments, more than 40 police departments and 80 school districts.

Los Angeles is extreme but not unusual. Alabama’s Jefferson County, which is one of seven counties in the Birmingham area, is home to 35 municipalities, including Cardiff, population 45. A map of the incorporated areas of Jefferson County (see map) clarifies matters somewhat: dozens of independent cities slot neatly into the gaps around Birmingham like a tricky jigsaw puzzle. From the air, it may appear that Birmingham sprawls but, in an administrative sense, the city is constrained. This is a problem. Cities that are unable easily to expand their boundaries are poorer, more segregated, have higher concentrations of poverty, lower growth, worse municipal bond ratings and less well-educated workforces.

America’s constitution goes to great lengths to demarcate state and federal powers. The Tenth Amendment explicitly reaffirms that “powers not delegated to the United States by the constitution, nor prohibited by it to the states, are reserved to the states respectively, or to the people.” Cities derive their power from state governments, says David Rusk, a former mayor of Albuquerque whose book on annexation, “Cities Without Suburbs”, is in its fourth edition. All but 11 states follow “Dillon’s Rule”, named after a 19th-century judge, which holds that municipalities only possess powers specifically granted to them by state law.

The joy of annex

For municipal officials, one way around this is to annex commercial territory, such as shopping malls, ports (as in LA ) or airports (Chicago), where there are few resident voters to irk and the tax returns are substantial. In the 1980s Birmingham annexed Overton. A shopping centre there, called the Summit, contributes about 15% of the city’s total revenues, says Darrell O’Quinn, a Birmingham city councillor. Some states, including Alabama, require cities to be contiguous, explaining the skinny tentacles that reach out to capture territory. The result is the higgledy-piggledy mess of urban boundaries.

Most states make it difficult for a city to force its neighbours to join it. Only eight states allow municipalities to annex territory unilaterally. Eight require the state legislature to change municipal boundaries. The overwhelming majority, 29 states, require a referendum in the areas to be annexed, according to an analysis by Greg Lindsey of the University of Minnesota. These votes often fail. Residents of unincorporated areas resist annexation, fearing a greater tax burden. Those in already rich suburbs fret about sharing their taxes with the poorer core city and merging of school districts. Municipalities avoid annexing poor neighbourhoods because they cost more to service than they provide in revenue. In places like Birmingham, blacks worry that rich, white suburbanites will usurp their hard-won power.

Divided and bonkers

The result is duplication and waste as municipalities each pay councillors, police and fire departments, waste-collection agencies and school administrators to perform the same services. Cities are reluctant to co-operate even on menial things like waste collection, fearing an erosion of their independence. Fragmentation is one of the main reasons that many cities are poor at providing public transport.

Businesses use fragmentation to their advantage, making known their interest in moving to a metropolitan area and playing its constituent cities off against each other. “We get into a bidding war trying to attract companies, and the only one who wins is these companies,” says Mr O’Quinn. Firms win tax breaks and other enticements; employees’ use of infrastructure puts a further burden on the whole conglomeration.

The jigsaws also deepen inequality. Small, wealthy cities can afford better schools. No matter how much young white couples enjoy living in restored downtown lofts and strolling to artisan coffee shops full of reclaimed wood, they eventually move to suburban houses in superior school districts when they have children, taking their taxes with them. Struggling cities get poorer; comfortable ones get richer. It is, for families, a sensible decision: four of the five best school districts in Alabama are in the suburbs of Birmingham, according to Niche, a website. Mountain Brook, a Birmingham suburb, is home to the highest-ranked school system in the state. Its median household income is four times that of the mother city. Its poverty rate is one-seventh.

Overcoming restrictive state laws requires creativity. In the 1960s Minneapolis, St Paul and their surrounding suburbs lost revenue as they competed to attract business. In 1971 Minnesota legislated that all municipalities in the region had to put part of their commercial taxes into one kitty. Today the Twin Cities area is one of America’s most successful, with a rare mix of well-off residents and affordable homes.

A more recent example is Charlotte, North Carolina, another southern city that shares some characteristics with Birmingham. North Carolina had some of the most liberal annexation laws in the country; until recently it allowed cities to annex contiguous land more or less unilaterally. Between 1950 and 2010, Charlotte’s area grew by 892%. Today it is the economic powerhouse of North Carolina. Its median income is higher by two-thirds, the unemployment rate is lower by a third and the poverty rate is almost half that of its Alabaman counterpart, according to a report by the Public Affairs Research Council of Alabama ( PARCA ), a non-partisan think-tank. Charlotte has a viable public-transport system and the corporate headquarters of Bank of America. It attracts young workers in droves.

There are other templates for cities hobbled by fragmentation. Pittsburgh is only one of Allegheny County’s 130 municipalities. But it spent decades working with the county and its many neighbours to foster regional co-operation, which is bearing fruit as it emerges as a tech hub. Louisville merged with Jefferson County in Kentucky, uniting its 92 neighbouring municipalities as “Greater Louisville”.